Optimization of Fiber-Reinforced Polymer Patches for Repairing Fatigue Cracks in Steel Plates Using a Genetic Algorithm
Publication: Journal of Composites for Construction
Volume 24, Issue 2
Abstract
A practical design optimization of fiber-reinforced polymer (FRP) patches for repairing fatigue cracks in metallic structures is presented. The design procedure combines finite-element (FE), genetic programming (GP), and genetic algorithm (GA) approaches. An optimum patch design is defined as the combination of design parameters that simultaneously minimizes the patch volume and reduces the stress intensity factor (SIF) range below the fatigue threshold range. A patching correction factor, which accounts for the positive effects of material and geometric properties of the patch and adhesive layer on the SIF solution, is proposed. The correction factor is developed by performing symbolic regression via GP analyses on the SIF values obtained from the three-dimensional FE models. The closed-form SIF solution facilitates the visualization of the effects of design parameters, simplifies the calculation of fatigue life, and reduces the computation effort for design optimization. An example of the center-cracked steel plate subjected to constant amplitude fatigue loading and then repaired with double-sided adhesive-bonded FRP patches is used to illustrate the optimization procedure.
Introduction
Due to their excellent mechanical properties, fiber-reinforced polymer (FRP) composites have been used to enhance the flexural strength (Miller et al. 2001; Lenwari et al. 2005, 2006; Zhao and Zhang 2007; Schnerch and Rizkalla 2008; Teng et al. 2012), lateral-torsional buckling capacity (Kabir and Seif 2011; Ghafoori and Motavalli 2015), local buckling strength (Zhao and Zhang 2007), and fatigue behavior (Colombi et al. 2003a; Liu et al. 2009a, b; Täljsten et al. 2009; Jiao et al. 2012) of steel structures. For cracked metallic structures, other fatigue crack repair methods, including the hole drilling, weld repair, use of doublers or splice plates, and posttensioning, have been used (Dexter and Ocel 2013). The adhesive-bonded FRP patches can significantly increase the fatigue life by reducing the crack tip stress intensity factor (SIF) (Liu et al. 2009a; Täljsten et al. 2009; Hmidan et al. 2014, 2015; Wang et al. 2016b; Zheng and Dawood 2017; Aljabar et al. 2017). A reduction in the SIF can be predicted by several calculation methods. Analytical works by Rose (1981, 1988) are applicable for cracked infinite orthotropic plates repaired with elliptical composite patches. The finite-element (FE) method has been the choice for more complex three-dimensional (3D) crack problems (Naboulsi and Mall 1996; Sun et al. 1996; Ayatollahi and Hashemi 2007; Lam et al. 2010; Gu et al. 2011; Wang et al. 2014).
Although many experimental and analytical works have investigated the effectiveness of bonded patches on enhancing the fatigue life of cracked metallic structures (Colombi et al. 2003b; Jones and Civjan 2003; Papanikos et al. 2007; Liu et al. 2009b; Nakamura et al. 2009; Lam et al. 2010; Gu et al. 2011; Yu et al. 2013; Wang et al. 2014; Albedah et al. 2015; Emdad and Al-Mahaidi 2015; Karatzas et al. 2015; Wang et al. 2016a; Zheng and Dawood 2016), there have been very few optimum design studies. Typically, an optimum design requires an objective function and constraints and aims to either minimize the cost function under constraints on the mechanical properties or maximize the mechanical property under some cost constraints. Kumar and Hakeem (2000) conducted a series of parametric FE studies to obtain the optimum patch shape that minimized the SIF of cracked aluminum sheets. Brighenti (2007) developed a tool in which a genetic algorithm (GA) was embedded in the FE code to obtain the patch topology that minimized the SIF of cracked steel plates. Errouane et al. (2014) combined the FE method, response surface methodology (RSM), and gradient-based optimization to minimize the composite patch volume for cracked aluminum sheets. Recently, Rasane et al. (2017) used RSM to minimize the patch area for center-cracked aluminum sheets. However, the previous optimization designs involve a complicated iterative process.
In this paper, a practical design optimization of bonded FRP patches for repairing fatigue cracks in steel plates is described. The patch volume represents the structural cost, while the SIF represents the mechanical property. Optimum patch design refers to a combination of design parameters that simultaneously minimizes the patch volume and reduces the SIF range below the fatigue threshold range. Therefore, further crack propagation is prevented. Once an optimum patch is obtained, the patch area associated with the costs required for surface preparation and adhesive bonding can be evaluated.
Fig. 1 shows the proposed design optimization procedure. The procedure combines FE, genetic programming (GP), and genetic algorithm (GA) approaches. A database of SIF values is generated from the 3D FE analysis. Then, symbolic regression via GP is performed on the SIF database to develop a closed-form SIF solution. The GP is performed in a HeuristicLab environment (Wagner et al. 2014). The SIF solution facilitates visualization of the effects of design parameters on the SIF and fatigue life calculation. The closed-formed SIF solution also replaces the iterative SIF analyses required during design optimization. Therefore, the computation effort required for the design optimization is reduced. In this study, GA is chosen as the optimization solver because it is suitable for an optimum problem with complex design variables (Fitzpatrick and Grefenstette 1988). The optimum design is also checked to determine the possibility of patch rupture and debonding failures. The limits of design parameters can be extended when the optimum solution does not pass the patch rupture or debonding failure criterion.
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An example of the center-cracked steel plate subjected to constant amplitude fatigue loading and then repaired with double-sided adhesive-bonded FRP patches is used to illustrate the design optimization procedure. A total of 864 FE models were analyzed to create the SIF database for the FRP-patched cracked plates. The performance of GA was also compared with nonlinear programming in MATLAB version 2018a (MathWorks 2018).
Stress Intensity Factor Solution
According to Tada et al. (2000), a theoretical SIF solution for a finite-width center-cracked plate subjected to a remote tensile stress is given in Eq. (1)
(1)
The correction factor () is given in Eq. (2)where = stress intensity factor; = correction factor for a finite-width plate; = one-half of crack length; = plate width; and = remote tensile stress.
(2)
To account for the positive effects of the material and geometric properties of the patch and adhesive layer on the SIF of FRP-patched cracked plates, a so-called patching correction factor () is proposed, and so Eq. (1) becomes Eq. (3)
(3)
The terms to are defined as shown in (4)where = patching correction factor; = elastic modulus of steel; = longitudinal elastic modulus of patch; = elastic modulus of adhesive; and = width and length of patch, respectively; and the terms , , and are the thickness of the steel plate, patch, and adhesive layer, respectively. The development of the function will be described in subsequent sections. An additional correction factor ( function) may be added to the right-hand side of Eq. (3) to account for debonding effects in future works. However, in this study, Eq. (3) is used in an inequality constraint for minimization of the patch volume.
(4)
Three-Dimensional Finite-Element Models
Elements and Discretization
A total of 864 3D FE models were analyzed with the ABAQUS software version 2017 to obtain the SIF values for different combinations of steel plate configurations, crack lengths, materials and the geometric properties of the patch, and the adhesive layer. A 20-node quadratic solid element, C3D20, was used for both the cracked plate and adhesive layer. An 8-node continuum shell element with reduced integration, SC8R, was appointed to the FRP patch. The layer-wise theory of Reddy (1987) was applied for modeling the FRP patch by stacking SC8R elements throughout its thickness using the command *SHELL SECTION, STACK DIRECTION = 3. To capture the square root singularity of the stress and strain fields near the crack tip, the authors assigned a collapsed 3D element to elements in a small region around the crack front.
Surrounding the crack front are strips of wedge-shaped elements. These elements fill a semicylinder, radius, , with a center at the crack front. The number of strips spanning along the radial direction of the semicylinder depends on the crack front element size, . To determine an appropriate value of , a sensitivity analysis (Hmidan et al. 2015) was implemented on the center-cracked steel plate. In the sensitivity analysis, the calculated SIF values, , were obtained from different ratios (from 20 to 100) and two normalized crack lengths ( and 0.9). Fig. 2 shows a comparison between the calculated SIFs and referenced solutions, (Tada et al. 2000). For both normalized crack lengths, the ratio converged to one as the ratio approached 100, and so ratio of 100 was chosen. The element size of was used for the patch, adhesive layer, and steel plate region below the adhesive layer, while the element size of was used in other regions of the steel plate.
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Materials, Geometries, and Constraints
Table 1 shows the material and geometric properties of steel plates, FRP patches, and adhesive layers used in FE models. Four different steel plates were used for two different purposes. Plates 1 () and 2 () were employed to create a database for developing the patching correction factor, , while Plates 3 () and 4 () were used to obtain independent results for verification of the developed SIF solution. The elastic modulus (), Poisson’s ratio (), and fatigue threshold SIF range of steel were assumed to be 200 GPa, 0.3, and 6.6 MPa (taken at a stress ratio of 0.13), based upon Pook (1975), respectively. Three unidirectional FRP materials of different longitudinal () and transverse () elastic moduli [Sika CarboDur M and H (SikaAG 2009) and MBRACE 460/1500 (Wu et al. 2012)] were chosen. Three adhesive materials [FM-73, FM36, and FM400 (Duong and Wang 2010)] of different elastic modulus () and Poisson’s ratio () values were chosen. The steel and adhesive materials were assumed to be isotropic and linearly elastic, while the FRP materials were linear orthotropic under a plane stress condition. The FE models assumed a linear elastic behavior, which is reasonable for fatigue loading conditions.
Part | Elastic modulus (GPa) | Shear modulus (GPa) | Poisson’s ratio | Thickness (mm) | Width (mm) | Length (mm) | |
---|---|---|---|---|---|---|---|
Steel plate | |||||||
Plate 1 | 200 | 77 | 0.30 | 6.0 | 90 | 1,300 | |
Plate 2 | 200 | 77 | 0.30 | 16.0 | 180 | 2,600 | |
Plate 3 | 200 | 77 | 0.30 | 10.0 | 100 | 1,500 | |
Plate 4 | 200 | 77 | 0.30 | 12.0 | 150 | 2,200 | |
FRP patch | |||||||
Sika CarboDur Ma | 210 | 8 | 5 | 0.30 | 1.2 | ||
Sika CarboDur Ha | 300 | 12 | 5 | 0.30 | 1.4 | ||
MBRACE 460/1500b | 460 | 12 | 5 | 0.36 | 2.0 | ||
Adhesive layer | |||||||
FM-73c | 959 | 355 | 0.35 | 1.0 | |||
FM36c | 1,815 | 672 | 0.35 | 1.0 | |||
FM400c | 2,944 | 1,082 | 0.36 | 1.0 |
Fig. 3 shows a typical quarter model of the center-cracked steel plate repaired with adhesive-bonded double-sided FRP patches. Displacement symmetric constraints (Xsym and Ysym) were applied to the corresponding symmetric planes. To enforce geometric compatibility conditions along the steel-adhesive and adhesive-patch interfaces, the authors employed the surface tie constraint. The tie constraint is a reasonable choice when the debonding effect is minimal, as observed in some experimental studies (Akbar et al. 2010; Wu et al. 2012; Yu et al. 2012; Wu et al. 2013).
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A matrix of 864 rows was constructed from the FE models of two steel plates (Plates 1 and 2), three normalized crack lengths (, 0.5, and 0.9), three patch materials, four normalized patch widths (, 0.6, 0.8, and 1), four normalized patch lengths (, 6, 11, and 16), and three adhesive materials [Table 1]. The length () and width () of the adhesive layer were equal to those of the patch. The calculated SIF value for each row occupied the last column of the matrix. The database was obtained by normalizing the calculated SIF values with the remote tensile stress, crack length, and finite-width correction factor.
Validation of FE Models
The FE models of FRP-patched cracked plates were validated using data from past analytical and experimental studies (Kumar and Hakeem 2000; Ayatollahi and Hashemi 2007; Lam et al. 2010). Fig. 4 shows that the obtained SIF values are in good agreement with the published results for both single and double patches. In Fig. 4(a), the FE models can capture the singular stress field in cracked aluminum plates repaired with single-sided patches (Ayatollahi and Hashemi 2007), with an average error of 1.6%. For the cracked aluminum plates repaired with double-sided patches (Kumar and Hakeem 2000), the average error was 1.9% [Fig. 4(b)]. As shown in Fig. 4(c), the FE model can reasonably predict axial strains in the vicinity of a crack tip for both patched and unpatched sides of the test specimen R116 in Lam et al. (2010).
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GA and GP Approaches
Use of GA
Genetic algorithms are iterative numerical solvers for optimization problems that were inspired by natural selection and natural genetics (Mitchell 1998). Each GA operates on a population of binary strings in the computer program. The GA starts by encoding random numeric values of independent variables in solution space to binary strings in the computer program with respect to the 1-to-1 mapping property. Each binary string represents one numeric point and vice versa. After the encoding process, the GA determines the fitness of each string in the current generation. The fitness is a value of the specified objective function at a particular point corresponding to the string under consideration. The algorithm then arranges all strings in a descending order of fitness values. Based on the order, GA sequentially performs the elite transfer, crossover, and mutation operators (Mitchell 1998) to produce a new population for the next generation. These operations are then repeated for each subsequent generation until the algorithm terminates at a specific number of generations.
Use of GP
The SIF of patch-repaired cracked plates depends on several design parameters. Conventional regression methods, which assume a mathematical model with unknown coefficients, are not efficient. A symbolic regression that simultaneously derives a mathematical model and unknown coefficients is preferred. In this study, GP is adopted as a tool for developing the SIF equation because it effectively minimizes residual errors between the source data and prediction. GP is an important application of GA for regression analysis (Koza 1992; Sette and Boullart 2001). A major difference between GP and GA is the representation of candidate solutions in which binary strings are used in GA and tree structures in GP (Fig. 5).
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In this study, GP provides the best fit mathematical model for the patching correction factor () in Eq. (3). A mathematical model that correlates the numerical values with the GP approximations is obtained by using GP to maximize the squared Pearson correlation coefficient (Pearson’s ) in Eq. (5) (Lee Rodgers and Nicewander 1988)where = number of values normalized from the SIF database (864 in this case); = , obtained from the th FE analysis; = mean of all values; = approximate value of (randomly generated by GP); and = mean of all values.
(5)
The GP analysis consists of the following steps. In the first step, the initial population with a specific number of tree structures is randomly created using atoms from a given function set () and terminal set (). Atoms of the set can be arithmetic operations, mathematical functions, Boolean operations, conditional operators, or any user-defined function (Koza 1992). Within a tree structure, atoms occupy functional nodes that have two arguments, as shown in Fig. 5. The atoms are independent variables and constants (Koza 1992) that are located at terminal nodes with no argument. In the second step, GP determines the values for all existing tree structures in the current generation. Based on the values, the algorithm stores all tree structures in a column vector and arranges them in descending order of their values. To produce a new population of tree structures for the next generation, GP sequentially implements three genetic operators: elite transfer [Fig. 6(a)], crossover [Fig. 6(b)], and mutation [Fig. 6(c)]. As a GA’s family member, GP’s solution is determined as the best-so-far tree structure stored in the cache of the computer program. The solution can be improved with a longer running time of the algorithm. For GP analysis, this study uses the HeuristicLab environment developed by Wagner et al. (2014). Different GP analyses can provide different SIF solutions. However, all solutions are capable of predicting SIF values if they are derived from the same database and GP’s input (Do and Lenwari 2018).
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Closed-Form SIF Solution for FRP-Patched Cracked Plates
GP Analyses
In the HeuristicLab, the database was randomly shuffled into four groups (1, 2, 3, and 4) to avoid any bias of the GP performance on certain groups. Each group had four subgroups (A, B, C, and D), each of which contained 216 data points (25% of the database) (Table 2). Subgroups A, B, C, and D were then sequentially used as a test set of 25% of the database, while in each case, the other three subgroups (75% of the database) formed the training set. Thus, four different functions were obtained from the four data groups, and the function with the highest value was chosen.
Group | Training set | Test set | ||
---|---|---|---|---|
Subgroup | Number of data points | Subgroup | Number of data points | |
1 | A, B, C | 648 | D | 216 |
2 | A, B, D | 648 | C | 216 |
3 | A, C, D | 648 | B | 216 |
4 | B, C, D | 648 | A | 216 |
The function and terminal sets were and , while the independent variable atoms were , , , and , as given in Eq. (4). Constant atoms were initially generated within an interval ]. Additional control parameters defined at the beginning of each GP analysis are listed in Table 3. The termination for all GP analyses was at the 2000th generation. All GP analyses were performed using an Intel Core i7-7700HQ CPU @ 2.81 GHz.
Parameter | Value |
---|---|
Number of tree structures | 10,000 |
Probability of mutation | 25% |
Elite count (reproduction option) | 2 |
Maximum number of tree depth | 10 |
Maximum number of tree length | 30 |
Fig. 7 shows that the training Pearson’s for the four GP analyses was either improved or remained constant after each GP generation. At the 2000th generation, the training values corresponding to the four groups were 0.902, 0.901, 0.900, and 0.906, respectively. The value of higher than 0.9 was an indicator of a good correlation for the training set. Fig. 8 shows the scatter diagram of at the 2000th generation in the case of (Group 4) for training data ( for test data). The mean absolute error for the training data was 0.059. Although the FE and GP results were correlated, a discrepancy between the FE and GP results was observed at high values, when a crack is not entirely covered with the patch. Therefore, the SIF solution is recommended for complete crack coverage.
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Patching Correction Factor () Function
Fig. 9 shows the chosen GP’s tree structure with training for the patching correction factor () in Eq. (3). The corresponding mathematical function is given in Eq. (6)where , , , and are the normalized parameters as defined in Eq. (4); and are the constant coefficients, as shown in Fig. 9. The application range of Eq. (6) is shown in Table 4. When the design parameters are beyond the range of applicability in Table 4, the FE and GP analyses in accordance with the previous sections can be performed to generate the new correction factor function.
(6)
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Part | Parameter | Unit | Min | Max |
---|---|---|---|---|
Steel plate | GPa | 200 | 200 | |
mm | 90 | 180 | ||
mm | 6 | 16 | ||
— | 0.1 | 0.9 | ||
FRP patch | GPa | 210 | 460 | |
GPa | 8 | 12 | ||
— | 0.2 | 1.0 | ||
— | 1 | 16 | ||
mm | 1.2 | 2.0 | ||
Adhesive layer | MPa | 959 | 2,944 | |
mm | 1 | 1 |
Verification of SIF Solution
Independent FE analyses and fatigue experiments in the literature were used to verify the proposed SIF solution. Plates 3 and 4 with four combinations of crack lengths, material and geometric properties of the FRP patch, and an adhesive layer were used in the FE analyses. Fig. 10 shows a comparison of the factors predicted by Eq. (6) with the independent FE results, where the predicted values were in good agreement with the FE results. For Plate 3, the average differences between the GP and FE solutions for different crack lengths, patch lengths, and patch widths at or 0.9 were 3%, 5%, 3%, and 5%, respectively. For Plate 4, the differences were 2%, 4%, 2%, and 7%, respectively. However, the proposed SIF solution became less accurate when a crack was not entirely covered with the patch, and a large discrepancy was observed at a crack length of , as shown in Fig. 10(d).
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In addition, the fatigue life of four specimens in the previous experimental studies was estimated using the proposed SIF solution. Fig. 11 shows a comparison between the predicted and measured fatigue crack growth curves. Specimens CS-1 and CS-2 (stress ) were tested by Wang et al. (2016b), while specimens D1L and D2L (stress range ) were tested by Emdad and Al-Mahaidi (2015). The Paris law constants used in the fatigue life calculation were (for MPa and m units), and (Maiti 2015). The fatigue life of a specimen with a crack growing from an initial crack length, , to a critical one, , was predicted by Eq. (7)where = effective range of SIF that accounts for crack closure effects (MPa ); and was assumed to be equal to .
(7)
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As shown in Fig. 11, a reasonable agreement was observed when the debonding effect was minimal. Therefore, the proposed SIF solution is applicable for predicting the fatigue life of FRP-patched cracked steel plates. The failure mode of specimen D1L was patch rupture, while that for specimen D2L was debonding. Debonding of the FRP sheet surrounding the fatigue crack was reported for both specimens CS-1 and CS-2, where the debonding area increased with an increasing number of cycles, but the CFRP sheets did not separate completely from the steel plates when the specimens fractured.
Patch Volume Optimization
Optimization Problem Statement
When the crack length, FRP and adhesive material properties, and adhesive thickness are specified, the correction factor in Eq. (6) will be a function of the FRP patch geometry. The SIF in Eq. (3) is rewritten as Eq. (8)where the terms to are derived as shown in Eq. (9)
(8)
(9)
The optimization statement minimizes the patch volume, , with respect to the constraint , where is the fatigue threshold SIF range of steel. The vector of design parameters is within the lower () and upper () bounds shown in Eq. (10)
(10)
In Eq. (10), the lower bound of the patch length is , which is the lower bound used in the GP analysis. It is recommended that the effective bond length be used as the lower bound in the optimization problem. The effective bond length is a function of the thickness and elastic modulus of the steel plate, FRP patch, and adhesive layer (Nozaka et al. 2005; Bocciarelli et al. 2007; Duong and Wang 2010).
Fig. 12(a) shows three surfaces corresponding to (Surf 1), (Surf 2), and fatigue threshold SIF range (Surf 3), respectively. The surfaces can help visualize the inequality constraint for the SIF range. When the patch thickness increases, SIF decreases. Therefore, the solution space, i.e., the underthreshold area, expands from the smallest space [Fig. 12(b)] to the largest one [Fig. 12(c)] as the patch thickness increases from 1.2 to 2 mm. If the crack length increases, Surf 3 may lie under Surf 1 and Surf 2, and no feasible solution can be obtained.
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Optimization Solvers
The performance of GA (command ga) was compared with nonlinear programming (command fmincon) in MATLAB version 2018a (MathWorks 2018). Both optimization solvers do not warrant the global minima. For the purpose of direct comparison, both solvers used an identical input of an initial point of the upper bound, function tolerance of , constraint tolerance of , and a maximum number of generations of 100.
Design Example
Problem Definition
A center-cracked steel plate (, , and ) was considered in the design example. The elastic modulus () and Poisson’s ratio () of steel were 200 GPa and 0.3, respectively. The fatigue threshold SIF range () was 6.6 MPa or 209 MPa , (Pook 1975). The steel plate was subjected to constant amplitude cyclic loading with a tensile stress range of ( and ), and two normalized crack lengths of and were considered.
Design Optimization Solutions
A total of nine combinations of patch and adhesive materials were used to investigate the effects of the material properties on the optimum solutions at two different crack lengths. For each combination, GA was performed twice to examine its stochastic property.
Based on the lower and upper bounds of the design parameters, the GA solver randomly generated a total of 500 binary strings in which each string had a particular fitness value (patch volume value). Therefore, there were 500 different fitness values at each GA generation.
Fig. 13 shows the convergence histories of the 1st and 2nd GA, and the fmincon analyses for the case when , , and . The best and mean fitness refers to the lowest and mean values of 500 fitnesses, respectively. Although the mean fitness values at the beginning of both analyses were much higher than the optimum ones, the GA could quickly achieve the solutions by the 8th and 7th generation for the 1st and 2nd GA analyses, respectively. The difference between the results of both GA analyses was 0.5%. Meanwhile, fmincon required 23 iterations to achieve a better solution (0.6% lower than the GA solutions).
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In the case of a longer crack (, , and ), the same conclusions could be drawn. Fig. 14 shows that the way GA searched for the optimum solution was smoother than fmincon. Again, fmincon required more iterations to achieve the optimum solution, which was 2.2% lower than the GA solutions. Note that although Figs. 13 and 14 show that the patch volumes obtained from fmincon were the lowest after the 7th and 2nd iteration, respectively, these are not the minima because the first Karush-Kuhn-Tucker condition (Rao and Rao 2009) was violated.
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Table 5 summarizes the optimum patch solutions for different patch and adhesive materials at two normalized crack lengths. It was found that the GA solution provided an equal or slightly higher patch volume than the fmincon solution. The maximum differences between the two solvers were 0.6% and 2.2% for values of 0.2 and 0.3 , respectively. Also, the GA’s stochastic property was not pronounced, and the difference between the two GA solutions was trivial.
Design input | Optimum solution | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1st GA | 2nd GA | fmincon | ||||||||||||||
(GPa) | (MPa) | (mm) | (mm) | (mm) | () | (mm) | (mm) | (mm) | () | (mm) | (mm) | (mm) | () | |||
0.2 | 210 | 959 | 82 | 108 | 1.2 | 10,616 | 80 | 111 | 1.2 | 10,610 | 81 | 110 | 1.2 | 10,607 | 0.9 | 6.1 |
210 | 1,815 | 80 | 110 | 1.2 | 10,590 | 81 | 110 | 1.2 | 10,589 | 81 | 110 | 1.2 | 10,589 | 0.9 | 6.1 | |
210 | 2,944 | 81 | 109 | 1.2 | 10,566 | 80 | 110 | 1.2 | 10,621 | 80 | 109 | 1.2 | 10,565 | 0.9 | 6.1 | |
300 | 959 | 72 | 100 | 1.2 | 8,660 | 73 | 99 | 1.2 | 8,655 | 74 | 97 | 1.2 | 8,654 | 0.8 | 5.4 | |
300 | 1,815 | 72 | 100 | 1.2 | 8,654 | 73 | 99 | 1.2 | 8,643 | 74 | 98 | 1.2 | 8,641 | 0.8 | 5.4 | |
300 | 2,944 | 74 | 98 | 1.2 | 8,626 | 74 | 97 | 1.2 | 8,627 | 74 | 98 | 1.2 | 8,624 | 0.8 | 5.4 | |
460 | 959 | 67 | 78 | 1.2 | 6,373 | 66 | 80 | 1.2 | 6,362 | 66 | 80 | 1.2 | 6,355 | 0.7 | 4.4 | |
460 | 1,815 | 66 | 78 | 1.2 | 6,362 | 66 | 80 | 1.2 | 6,348 | 66 | 80 | 1.2 | 6,347 | 0.7 | 4.4 | |
460 | 2,944 | 66 | 80 | 1.2 | 6,347 | 66 | 78 | 1.2 | 6,377 | 66 | 80 | 1.2 | 6,337 | 0.7 | 4.4 | |
0.3 | 210 | 959 | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
210 | 1,815 | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | |
210 | 2,944 | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | |
300 | 959 | 90 | 183 | 2.0 | 32,674 | 90 | 178 | 2.0 | 31,965 | 90 | 178 | 2.0 | 31,961 | 1.0 | 6.6 | |
300 | 1,815 | 90 | 177 | 2.0 | 31,815 | 90 | 177 | 2.0 | 31,811 | 90 | 177 | 2.0 | 31,811 | 1.0 | 6.6 | |
300 | 2,944 | 90 | 176 | 2.0 | 31,634 | 90 | 176 | 2.0 | 31,616 | 90 | 176 | 2.0 | 31,616 | 1.0 | 6.5 | |
460 | 959 | 86 | 91 | 2.0 | 15,613 | 86 | 91 | 2.0 | 15,614 | 87 | 90 | 2.0 | 15,611 | 1.0 | 3.3 | |
460 | 1,815 | 86 | 90 | 2.0 | 15,594 | 86 | 91 | 2.0 | 15,596 | 87 | 90 | 2.0 | 15,593 | 1.0 | 3.3 | |
460 | 2,944 | 87 | 90 | 2.0 | 15,568 | 85 | 91 | 2.0 | 15,577 | 87 | 89 | 2.0 | 15,568 | 1.0 | 3.3 |
Note: N/A = not available.
In the case of a shorter crack (), the optimum patch width varied from 0.7- to 0.9-fold the width of the steel plate, while the optimum patch length varied from 4.4- to 6.1-fold the crack length. The optimum patch thickness was 1.2 mm. The optimum patch volume was when was 210 GPa. It decreased by 18.4% and 40.1% when were 300 and 460 GPa, respectively. The lowest patch volume () was found when was 460 GPa and was 2,944 MPa. In addition, the optimum patch volume was found to be relatively insensitive to the adhesive modulus. The optimum patch volume slightly decreased from 0.1% to 0.2% when increased from 89% (from 959 to 1,815 MPa) to 207% (from 959 to 2,944 MPa).
In the case of a longer crack (), no feasible solution existed when was 210 GPa because the threshold surface Surf 3 lay below Surf 1 and Surf 2 (Fig. 12). However, an optimum patch volume could be achieved if the patch modulus increased from 210 to 300 or 460 GPa. As increased from 300 to 460 GPa, the patch volume decreased about two-fold. Similarly, the effect of the adhesive modulus on the optimum patch volume was negligible.
Assessment of Patch Rupture and Debonding Failures
To assess the possibility of patch rupture and debonding failures, the authors analyzed the optimum patch-repaired cracked plates in Table 5 at the maximum fatigue load. The rupture failure of each FRP patch was assessed using the Tsai-Hill failure criterion (Staab 2015), as shown in Eq. (11)where = Tsai-Hill failure index; , , and = longitudinal, transverse, and shear stresses in the patch, respectively; and , , and = ultimate longitudinal, transverse, and shear strength of the patch, respectively. The CFRP material strength in each direction is given in Table 6. The FRP rupture was deemed to have occurred when Eq. (11) was satisfied.
(11)
In recent years, the cohesive zone modeling approach has been employed to analyze adhesive-bonded joints, especially when the debonding behavior is simulated (Chen and Qiao 2009; Campilho et al. 2011, 2013; Chaves et al. 2014; Colombi et al. 2015; Zheng and Dawood 2016; Budhe et al. 2017). Typically, test data are required for cohesive parameters used in the traction-separation law. Due to its simplicity, the strength-based criterion was chosen to assess the debonding failure in this study. This criterion has been used in the assessment under static and fatigue loading conditions (Papanikos et al. 2005; Bocciarelli et al. 2009). However, the prediction can be sensitive to the element size due to the singularity condition. Small element sizes are required when this criterion is used (Da Silva et al. 2011). The debonding failure was assumed when at least one of the following three conditions was satisfied (Papanikos et al. 2005). For the first condition, the failure in the adhesive layer based on the maximum shearing stress at the steel-adhesive interface occurred when the Tresca reached a maximum, determined from Eq. (12)
(12)
For the second condition, the debonding at the steel-adhesive interface, based on the maximum normal stress, occurred when Eq. (13) was satisfied
(13)
For the third condition, the debonding at the adhesive-patch interface, based on the maximum normal stress, occurred when Eq. (14) was satisfied
(14)
In Eqs. (12)–(14), the terms , , and = adhesive failure indexes; and = maximum and minimum principal stresses, respectively; = normal stress (peeling stress); and , , and = shear strength of adhesive material, peeling strength of steel-adhesive interface, and peeling strength of adhesive-patch interface, respectively. The values for and (Table 6) were taken from Deng et al. (2018). When was assumed, was higher than because the maximum normal stresses at the steel-adhesive interface obtained from the FE analyses were higher than ones at the adhesive-patch interface.
Table 7 summarizes the patch and debonding failure indexes for the optimum patch solutions. Because all failure indexes (, , and ) were less than one, the FRP rupture and debonding failures were not possible for optimum patch solutions at the maximum fatigue load. The Tsai-Hill failure indexes () for patch rupture were low in most cases, while the Tresca index for the adhesive layer [Eq. (12)] became pronounced as the adhesive modulus increased [ for and 0.64 for , respectively]. In this example, the values were higher than values. If the loading magnitude increased, debonding failure at the adhesive layer could be possible. For the optimum patch solutions, a good correlation between the SIF ranges obtained from ABAQUS and Eq. (3) was observed.
(GPa) | (GPa) | (MPa) | SIF-Eq. (3) (MPa ) | SIF-ABAQUS (MPa ) | () Eq. (11) | () Eq. (12) | () Eq. (13) | |
---|---|---|---|---|---|---|---|---|
0.2 | 210 | 8 | 959 | 209.0 | 214.4 | 1.6 | 216.4 | 74.1 |
210 | 8 | 1,815 | 209.0 | 208.3 | 2.0 | 382.6 | 132.9 | |
210 | 8 | 2,944 | 209.0 | 199.5 | 1.0 | 581.4 | 219.1 | |
300 | 12 | 959 | 209.0 | 212.3 | 2.7 | 213.3 | 72.4 | |
300 | 12 | 1,815 | 209.0 | 201.0 | 2.3 | 369.0 | 127.1 | |
300 | 12 | 2,944 | 208.9 | 190.6 | 1.9 | 551.9 | 206.3 | |
460 | 12 | 959 | 209.0 | 213.2 | 1.7 | 216.2 | 72.6 | |
460 | 12 | 1,815 | 209.0 | 198.5 | 2.6 | 364.2 | 123.9 | |
460 | 12 | 2,944 | 209.0 | 190.2 | 3.0 | 529.8 | 206.7 | |
0.3 | 210 | 8 | 959 | N/A | N/A | N/A | N/A | N/A |
210 | 8 | 1,815 | N/A | N/A | N/A | N/A | N/A | |
210 | 8 | 2,944 | N/A | N/A | N/A | N/A | N/A | |
300 | 12 | 959 | 209.0 | 211.2 | 1.8 | 259.3 | 72.8 | |
300 | 12 | 1,815 | 209.0 | 198.8 | 1.7 | 434.8 | 123.7 | |
300 | 12 | 2,944 | 209.0 | 197.6 | 0.6 | 636.8 | 197.4 | |
460 | 12 | 959 | 209.0 | 217.5 | 1.7 | 300.1 | 82.9 | |
460 | 12 | 1,815 | 209.0 | 206.6 | 2.2 | 470.3 | 131.5 | |
460 | 12 | 2,944 | 209.0 | 195.8 | 2.7 | 643.2 | 196.0 |
Note: N/A = not available.
Figs. 15 and 16 show the effects of the adhesive modulus on Tresca and interfacial stresses in the adhesive layer for the two normalized crack lengths of and , respectively. The maximum Tresca, shear, and normal stresses in the adhesive layer increased as the adhesive modulus increased.
![](/cms/10.1061/(ASCE)CC.1943-5614.0001005/asset/8ce31b19-a9aa-4a88-9d69-8874dde846aa/assets/images/large/figure15.jpg)
![](/cms/10.1061/(ASCE)CC.1943-5614.0001005/asset/c12dd323-1cca-4c31-8bb0-d808ce5a0abc/assets/images/large/figure16.jpg)
For this design example, the use of a high modulus patch with a low modulus adhesive was recommended. A high modulus patch caused more load to be transmitted to the patch, while it did not significantly influence the maximum Tresca in the adhesive layer (Table 7). Meanwhile, a low modulus adhesive reduced the possibility of debonding.
Conclusions
The optimization of adhesive-bonded FRP patches for repairing fatigue cracks in steel plates is proposed. The design optimization combines FE, GP, and GA approaches. An optimum patch design is defined as a combination of design parameters that simultaneously minimizes the patch volume and reduces the SIF range at the crack tip below the fatigue threshold range. The FE models of the patch-repaired cracked plates from different combinations of design parameters are analyzed to obtain the SIF database. Based on the database, the symbolic regression via GP analysis is then implemented to develop a closed-form SIF solution, which is used in an inequality constraint. An example design is used to illustrate the design optimization procedure. The main conclusions are as follows:
•
Finite-element models with continuum shell elements stacked through the patch thickness and tie constraints representing geometric compatibility conditions along the steel-adhesive and adhesive-patch interfaces can capture a singular stress field in the vicinity of a crack tip in cracked steel plates repaired with bonded FRP patches.
•
Symbolic regression via GP provides a nonlinear mathematical function with sufficient accuracy for predicting the SIF values of patch-repaired cracked plates. The GP can be applied to other problems to derive an empirical model for structural behavior in terms of design parameters.
•
A patching correction factor is proposed to account for the positive effects of the material and geometric properties of the patch and adhesive layer. The closed-form SIF solution facilitates the visualization of the effects of design parameters, simplifies the calculation of fatigue life, and reduces the computation effort for design optimization.
•
For center-cracked steel plates under cyclic tension, the use of a high modulus patch with a low modulus adhesive is recommended for fatigue crack repair.
Data Availability Statement
Some or all data, models, or code generated or used during the study are available from the corresponding author by request.
Acknowledgments
The authors would like to acknowledge the financial supports from the ASEAN University Network/Southeast Asia Engineering Education Development Network (AUN/SEED-Net) and Chulalongkorn University.
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This work is made available under the terms of the Creative Commons Attribution 4.0 International license, http://creativecommons.org/licenses/by/4.0/.
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Received: Sep 23, 2018
Accepted: Aug 20, 2019
Published online: Jan 28, 2020
Published in print: Apr 1, 2020
Discussion open until: Jun 28, 2020
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