Kader MA^{*}, Kitane Y and Itoh Y  
Nagoya University, Nagoya, Aichi, JAPAN  
*Corresponding Author :  Kader MA Nagoya University, Nagoya Aichi, JAPAN Tel: +81528776988 E mail: [email protected] nagoyau.ac.jp 
Received May 25, 2015; Accepted June 25, 2015; Published July 05, 2015  
Citation: Kader MA, Kitane Y, Itoh Y (2015) Effect of BoltHole Position Errors on the Strength of BearingType MultiRow Bolted Connection of FRP Composite Members. J Civil Environ Eng 6:239. doi:10.4172/2165784X.1000239  
Copyright: © 2015 Kader MA, et al. This is an openaccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 
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The aim of the study is to evaluate the effect of bolthole position errors on the strength of bearingtype multirow bolted connections with a doublelap configuration of fiber reinforced polymer composite members. To simulate errors in positioning boltholes in the loading direction of connections, the statistical technique of sampling known as Latin hypercube sampling has been adopted in this study. In the Latin hypercube sampling, the distribution is obtained using the results of goodness of fit tests. It has been seen from the goodness of fit tests that the strength of the two to fourrow bolted connections with steel cover plates and tworow bolted connection with FRP cover Plates can be modeled by using the normal distribution and the three and fourrow bolted connections with FRP cover plates can be modeled by using the Weibull distribution with a significance level of 0.05. The strength of bolted connections is obtained by performing progressive damage analysis of the connections numerically. The statistical analyses show that the bolthole positioning errors have significant effect on the strength of the bolted connections, and that the strength can either increase or decrease with a larger probability to decrease. Connections with steel cover plates are more sensitive to bolthole position errors than those with FRP cover plates. Ninety five percent nonexceedance strength of the connection with FRP cover plates is about 85% of the connection with bolt positions perfectly aligned, that of the connection with steel cover plates is about 71% of the perfect bolt position case.
Keywords 
Bolthole position; FRP composite; Latin hypercube sampling; Bearingtype connection; Progressive damage analysis; Ultimate strength 
Introduction 
The civil engineering structures made of steel and concrete would deteriorate over time due to various reasons. In order to prevent the deterioration and reduce the lifecycle cost of a structure, the applications of fiber reinforced polymer (FRP) for a structure in a harsh environment may be one of the options. FRP composite materials do not corrode in the severe environment. The materials have other advantages in the context of structural applications such as high specific strength and stiffness, light weightiness, tailored properties, and outstanding fatigue resisting capacity [1]. The materials have been increasingly used in the civil engineering industry for strengthening and construction of structures for a few decades. 
Structural members need to be connected properly and adequately to build a reliable and safe structural system. In the civil engineering structure, mechanical fastener connections such as bolted connection provide more advantages than the other connections and are often chosen to connect members. However, a severe stress concentration occurs at bolt holes, and bolted connection of FRP composite members can be a source of weakness [2]. Furthermore, design guidelines of the bolted connection of the FRP members have not been fully developed yet. Therefore, many issues need to be resolved before this advanced material can be fully utilized to provide reliable and efficient connections. 
Bolted connections can be of either a frictioncritical type or a bearing type. Since strength of a frictioncritical bolted connection if FRP structures cannot be determined easily due to creep of the FRP material, bearingtype bolted connections are typically used. Strength of a bearingtype bolted connection depends on the failure mode. Common failure modes in a bolted connection of FRP structures are (a) bearing failure, (b) nettension failure, (c) shearout failure, (d) cleavage failure, (e) cleavagetension failure, and (f) bolt failure. In addition, failure may consist of their combination. Among these failure modes, the bearing failure mode is considered to be a desired mode because it is less catastrophic than other failure modes. 
Many researches have been conducted to determine the effect of various parameters on the strength of bolted connections of FRP members [212]. These parameters include: (a) connection geometry and plate width to bolt diameter ratio, w/d, pitch distance to bolt diameter ratio, p/d, and end distance to bolt diameter ratio, e/d; (b) connection configuration; (c) stacking sequence; (d) fiber orientation; (e) friction and bolt torque; (f) clearance of hole; and (g) cover plate stiffness. 
The present study focuses on the bearingtype multirow bolted connections with a doublelap configuration of FRP members. Bolts in different rows do not shear the same amount of force due to the relative displacement of the cover plates to the main plate. The load distribution depends on the several factors such as the relative stiffness of the cover plates to the main plate, bolthole position and clearance, axial force in the bolt, friction between FRP plates and between FRP plate and washer. For a ductile material, the load distribution among the bolt rows does not affect the ultimate strength of the connection since the load would be redistributed among the rows of bolts due to the plastic deformation [13]. However, the load distribution in a connection of a brittle material like FRP affects its ultimate strength because the load does not redistribute among the bolts. 
The authors [8,14] investigated the effect of the cover plate stiffness on the behavior of multirow bolted connection up to four rows of bolts by numerical analysis. The results show that the load distribution among the bolts changes with the change of cover plate’s stiffness and that a connection with larger stiffness of cover plate tends to show lower strength. MaCarthy et al. [7] and Fan et al. [15] investigated the load distribution among the bolts with bolthole position error. They investigated the threerow bolted connections with bolt holes perfectly positioned and positioned with an error of 80 μm, 160 μm, and 240 μm for FRP cover plates. They used a nominal hole diameter of 8 mm. The ISO specifies the tolerance of the f7/H10 fitting of 86 μm for 8 mm bolt hole which is in used by European aircraft manufacturer. They showed that the bolthole position error affects load distributions among the rows of bolts which may lead to early damage of bolt holes. 
The aim of this paper is to examine how bolthole position errors effect the strength of bearingtype multirow bolted connections with a doublelap configuration of FRP composite members. The bolthole position errors are randomly generated by Latin hypercube sampling. Steel and FRP cover plates with a half of the main plate thickness are used in the connections. Progressive damage analyses are performed on the connections with bolthole errors to understand the effect of bolthole position errors on the strength of the connections with steel and FRP cover plates. A statistical test is conducted on the results to obtain the type of statistical distribution. 
Materials and Methods 
Connection geometry 
The bearingtype multirow bolted connections a doublelap configuration with two, three, and four rows of bolts in a line are examined by finite element analysis, which are shown in Figure 1. The main plate is an FRP plate with a thickness, tm, of 12 mm. Two types of cover plate material are used: FRP and steel. The thickness of cover plates is 6 mm. Steel bolts with a diameter, d, of 16 mm and the bolthole diameter, dh, of 17 mm are used, resulting in a clearance of 1 mm. The clearance changes with the change of bolthole position. The bolthole position error is described in the next section. 
The geometric parameters used in the study are shown in Table 1. The plate width to bolt diameter ratio, w/d, is different for connection with a different number of bolts, but the pitch distance to bolt diameter ratio, p/d, and the edge distance to bolt diameter ratio, e/d, are fixed all connections. Based on the geometry considered in this study, the connections are expected to be bearing failure. 
Sampling 
To determine the effect of bolthole position errors on the strength of bolted connections, the Latin hypercube sampling technique is adopted in this study for simplicity and accuracy. The Latin hypercube sampling was first proposed by McKay et al. [16]. To do a special type of Monte Carlo simulation, Latin hypercube sampling uses the stratification of the theoretical probability distribution function of input random variables. At first, the cumulative probability distribution functions (CPDF) for all random variables are divided into N intervals of equal probability and then centroids of intervals are used in the simulation process. The samples can be obtained by using the formula 
(1) 
where x_{i,k} is the kth sample of the ith variable X_{i} , F_{i} ^{1} is the inverse CPDF for variable X_{i}. 
Based on random permutations of integers 1, 2, .... N, the representative parameters of variables is selected randomly. Every interval of each variable will be used only once during the simulation. In this study, fifty simulations are performed for each connection. Therefore, fifty different bolthole positions are generated by using the Latin hypercube sampling for each bolt hole and randomly selected for the bolt holes of a connection in which every interval of each bolthole position is used only once during the simulation. 
To determine the statistical distribution of the bolthole position error, 257 bolt holes in a number of connection GFRP plates to be used in the experimental study of bolted connections are measured. The measured data are found to follow a normal distribution with the mean of 0.014 mm and standard deviation of 0.099. The maximum bolthole position error is of 0.41 mm. In the study, the bolthole position errors are assumed to be normally distributed with a mean value of 0.00 mm and a standard deviation of 0.17 mm for all bolt holes. In the Latin hypercube sampling, 50 intervals are used. To include ± 0.4 mm in the sampling data, which is about the same magnitude as the maximum error found in the measurement, the standard deviation of distribution is set to 0.17 mm in place of 0.099. 
Material properties 
Quasiisotropic glassfiber laminates are used in this study. The stacking sequences of the 6mm cover plate and the 12mm main plate are [ 0° / 90° / ±45°]_{2S} and [0° / 90° / ±45°]_{4S} respectively. The thickness of each ply in the laminates is 0.375 mm. The material properties of unidirectional lamina are given in Table 2. Bolts and plates are of stainless steel, SUS 304, and their properties are shown in Table 3. Stressstrain curve of the stainless steel used in this study is the one specified by Eurocode 3. 
Finite element model 
Three dimensional finite element models of bolted connections are created in the general purpose finite element software, Abaqus [18]. A quarter of a connection is modeled by taking advantage of symmetric conditions as shown in Figure 2. At the center of the cover plate xsymmetric boundary conditions are defined, and a displacement is applied at the continuous edge of the main plate. Three dimensional solid eightnode elements are used to model FRP composite plates, steel plates, and steel bolts and washers. Washer and bolt are modelled together as a single part. The element size near the bolt hole about 0.75 mm × 1.5 mm × 3.0 mm, and that of the other portion is about 1.5 mm × 2.0 mm × 3.0 mm. The thickness of element is divided into 8 layers to model the stacking sequence. The element size was determined by examining several different meshes with different mesh sizes. The total number of elements in the model shown in Figure 2 is 20343 although it may vary depending on the analytical case. Surface based contact definition is employed where different parts may contact each other. In the contact definition, the Coulomb friction model with a frictional coefficient of 0.2 is used [19,20]. Fingertighten torque that is equivalent to an axial pretension force of 500 N is assumed and applied to the bolts. The position of bolt hole is changed only in cover plates. 
Progressive damage analysis 
The progressive damage analysis consists of stress analysis, failure criteria and material property degradation. Two principal failure criteria are considered: a) fiber failure and b) matrix failure. The failure criteria are expressed by Eqs. (2) to (4) [14]. 
a) Fiber failure criterion 
For fiber tension 
(2) 
For fiber compression () 
(3) 
where, XT is tensile strength and XC is compressive strength in the fiber direction, S_{ij} and G_{ij} are shear strength and shear modulus in the ij plane, respectively, α is a parameter representing the nonlinear relationship of the shear strain and shear stress, the value 1.9×10^{−6} is used, ii is the ij components of effective stress defined as , and τˆ_{j} is the ij components of effective shear stress defined as. d_{I} is the damage index with respect to the failure mode I. Detailed description can be found in Ref [14]. 
b) Matrix failure criterion 
(4) 
where 
and 
Y_{T} and Z_{T} denote tensile strengths, and Y_{C} and Z_{C} denote compressive strengths in the two perpendicular directions to fiber. 
Once a failure criterion reaches 1.0, further loading will cause degradation of material stiffness coefficients. The stiffness coefficients are reduced diagonal components of stiffness matrix as proposed by Matzenmiller et al. [21]. The stiffness coefficients are controlled by the fiber damage index and matrix damage index that have a value between zero to one. The stiffness properties are gradually decreased after the failure of the material to keep constant stress until 95% damage. The stiffness is then considered to be zero at the 95% of damage. This progressive damage model is implemented through a subroutine UMAT in Abaqus. The model is illustrated in Ref [14] in detail. 
Results and Discussion 
Progressive damage analysis is performed of the two, three, and fourrow bolted connections with different bolthole positions. The results are discussed in the following sections. 
Load distribution 
Figure 3 shows the load distribution among the bolts of the threerow bolted connections with steel and FRP cover plates where there is no error in bolthole positions. For the better understanding of the load distribution among the bolts, load of the bolts is shown by the thicker line and bolt load ratios are shown by the thinner line in the figure. The bolt load ratio is defined as a ratio of a bolt load, P_{b}, to the connection load, P. All bolts in the connections begin to carry load at the same time. In the connection with steel cover plates, every bolt carries different loads, where the first bolt takes a larger load than other bolts and the load that each transfer successively reduces with the increase the bolt number. However, in the connection with FRP cover plates, the first and last bolts carry the same load which is larger than that of the intermediate bolt. In this study, the closest and farthest bolts to the end of a main plate are designated as the last and first bolts, respectively. 
In the Figure 3, the load increasing rate of the largest bolt load in the connections within the elastic limit is indicated by using a dotted line. In the connection with steel cover plates, the first bolt has the larger load increasing rate and reaches earlier at the damage initiation stage than the other bolts and then the bolt starts shed the load to the others bolts. The damage initiation strength is defined as a load of the connection at which the load increasing rate of the largest bolt load reduces with the connection load due to the damage of the hole. As can be seen in Figure 3, the damage begins at the bolt load about 21 kN where cover plates are steel or FRP. The first bolt also fails at first in the connection. However, the first and last bolts have the larger load increasing rate then the intermediate bolt of the connection with FRP cover plates and reach at a time to those stages. The load of intermediate bolt is not larger different with the first and last bolts. It means that the load distribution among the bolts of the connection with FRP cover plates is more uniform than the connection with steel cover plates. Therefore, the connection with the FRP cover plates carries a larger load about 23% than that with steel cover plates [14]. The same results are also obtained in the two and fourrow bolted connections. It indicates that the strength of a connection is affected by the load distribution among the bolts. A connection will be tolerated the maximum load when the load distribution among the bolts is uniform. 
The load distribution among the bolts depends on the number of bolt rows, and the stiffness of the cover plate. However, when there are errors in bolthole positions, the load distribution among bolt rows is affected significantly. Figure 4 shows the load distribution among the bolts of the threerow bolted connection with position errors in the first and second holes of the cover plates along to the loading direction. 
The bolthole position error is illustrated by the Figure 5. It is observed that the last bolt of the connection contact with the cover plates and the main plate and begins to carry the load, and the other bolts are not in contact with the cover plates. The first and second bolts also begin to carry loads at the connection load of about 10 kN. After that the load increasing rates of bolt loads within the elastic limit of the material do not change, but the bolt load ratio is changed with the change of connection load. Therefore, the connection with steel cover plates, the first bolt picks up the largest load and reaches the damage initiation stage earlier than the other bolts. However, in the connection with FRP cover plates, the last bolt keeps the largest load and reaches the damage initiation stage first because the load increasing rates of the first and last bolts are the same after all bolts in contact with cover plates. It can also be observed that the connection with steel cover plates shows a larger ultimate load 8.8% when compared to the strength of that with bolthole positions perfectly aligned (Figures 3 and 4), although the ultimate strength of the connection with FRP cover plates is decreased to about 7%. It is because the load distribution among the bolts becomes more uniform for the connection with steel cover plates and less uniform for the connection with FRP cover plates due to the bolthole errors when compared that with a perfect bolt hole aligned. 
Based on the observation above, the strength of a connection increases when the bolt with a lower load increasing rate in the connection begins to carry load early and all the bolts reach almost the same time at the ultimate strength of the connection and failed at a time. On the other hand, the strength of a connection decreases when the bolt with a larger load increasing rate in the connection begins to carry load early and the bolts are failed in different interval of the connection load. 
Strength 
The damage initiation strength and ultimate strength of the connections are determined for each connection. These strengths of 50 connections with bolthole position errors are summarized in Table 4. Table 4 also shows the strengths of the connections with perfect alignment of bolt holes. 
Table 4 shows that the effect of bolthole error on the damage initiation strength is significant for the connection with either steel or FRP cover plates. In comparison, the variations of the strength are larger for the connections with steel cover plates than those with FRP cover plates. The coefficients of variation of the damage initiation strength for two, three, and fourrow bolted connections are 20.5, 25.0, and 26.7% for steel cover plates and 14.9, 18.9, and 18.7% for FRP cover plates, respectively. The average damage initiation strength is about 7% lower for steel cover plates and about 20% lower for FRP cover plates than that with bolthole position perfectly aligned. The connection with FRP cover plates shows the nonuniform load distribution a larger number of cases due to the error in the first and last bolt hole. Therefore, the average damage initiation strength is decreased more for the connection with FRP cover plates than that with steel cover plates. 
From Table 4, it can be observed that the connections with steel cover plates are more sensitive to the bolthole position errors than those with FRP cover plates. The ultimate strength of the two, three, and fourrow bolted connections with steel cover plates increases by about 8, 21, and 29% and decreases by about 46, 43, and 34%, respectively for the bolthole position errors. Whereas the connection with FRP cover plates of the tworow bolted connections, the strength does not increase, but decreases by about 14%. For the three and fourrow bolted connections, the strength increases by about 2.5 and 5% and decreases by 18 and 17%, respectively. The variation of the strengths is larger for the connections with steel cover plates than those with FRP cover plates. The coefficients of variation for the ultimate strengths are less than 6% for the connections with FRP cover plates. While they are greater than 13% for the connections with steel cover plates. The average ultimate strength is about 4% lower than that of a connection with bolthole position perfectly aligned for any type of connections examined in this study. 
To show the effect of bolthole position error on the ultimate strength of connections, F, the strengths are normalized by the strength of a connection with perfect bolthole position, F_{pbhp}. Figure 6 shows the frequency distribution of the normalized strength of the connections. The frequency is shown in terms of percentage. The interval of the normalized strength is set to 0.05. It can be observed that the strengths of the connections with steel cover plates have spread in a wider range than those with FRP cover plates. The frequencies are larger around the normalized strength of 1.0 for the connections with steel and FRP cover plates. It is because the bolthole position errors are normally distributed. 
Statistical test 
The statistical natures of the strength of bolted connections with different bolthole position are examined by using Latin hypercube sampling in the previous section. In this section, the distribution type and its statistical parameters are evaluated based on the results from the numerical analysis. For verifying the distribution type, for goodness of fit tests, namely KolmogorovSmirnov (KS) and Chi square (χ^{2}) tests are mostly used [22]. The advantage of the KS test over the χ^{2} test is that it is not necessary to divide the data into intervals, thus the error or judgment associated with the number of size of the interval is avoided. Therefore, the KS test is conducted in this study. In the tests for goodness of fit, normal, lognormal, and Weibull distributions are examined. 
Figure 7 shows CPDFs with the normalized strength of the twoto fourrow bolted connections with FRP and steel cover plates. The statistical identified parameters: mean, μ, standard deviation, σ, scale factor, λ and shape factor, ҡ, for those distributions are also shown in the figure. In the KS test, maximum deviations, D_{max}, of the CPDFs between the numerical results and the distributions are calculated and compared with the tabular value D_{α=0.05}. According to the KS test, the data is a good fit with the distribution when D_{max} is less than Dα .The goodness of fit tests shows that the strength of tworow bolted connections with steel and FRP cover plates and threerow bolted connections with steel cover plates does not fit with the Weibull distribution and also the strength of tworow bolted connections with steel cover plates does not fit with the lognormal distribution, but fit the other distributions with a significant level of 0.05. However, the strength of three and fourrow bolted connections with FRP cover plates and fourrow bolted connections with steel cover plates fit all of the three distributions with a significant level of 0.05. The statistical tests for goodness of fit shows that the strength of two to fourrow bolted connections with steel cover plates and tworow bolted connections with FRP cover plates can be better modeled by a normal distribution, and the strength of the three and fourrow bolted connections with FRP cover plates can be better modeled by a Weibull distribution with a significance level of 0.05. From the statistical models, it is found that 95% nonexceedance normalized strengths of the two, three and fourrow bolted connections with FRP cover plates are 0.88, 0.87, and 0.85 and those with the steel cover plates are 0.75, 0.71, and 0.74, respectively. 
Conclusions 
The effect of bolthole position errors on the strength of multirow bolted connections is investigated numerically. The bolthole position errors are considered in the bolt hole of cover plates to the loading direction. The bolthole position errors are assumed to be normally distributed and samples are generated by using Latin hypercube sampling. The following conclusions are made based on the numerical results. 
(1) In bearingtype multirow bolted connections, the load distribution among the bolts is not uniform for the connection with bolthole position perfectly aligned. The load distribution among the bolts is affected by the bolthole position errors. The bolthole position error will make a connection that the load distribution among the bolts will be either more or less uniform at the damage initiation strength or ultimate strength of the connection than that of bolthole position perfectly aligned. The load distribution will be more uniform when bolthole clearances in the first and last rows are larger than that of the intermediate row(s) for the connection with FRP cover plates. The load distribution of the connection with steel cover plates will be more uniform when the bolthole clearances in the intermediate and the first rows are larger than that of the last row. 
(2) The effect of bolthole error on the damage initiation strength is significant for the connection with either steel or FRP cover plates. However, variations of the damage initiation strength are larger for the connections with steel cover plates than those with FRP cover plates. The coefficients of variation of the damage initiation strength for two, three, and fourrow bolted connections are 20.5, 25.0, and 26.7% for steel cover plates and 14.9, 18.9, and 18.7% for FRP cover plates, respectively. The average damage initiation strength is about 7% lower for steel cover plates and about 20% lower for FRP cover plates than that with bolthole position perfectly aligned. 
(3) The ultimate strength of bolted connections is also affected by the bolthole position errors. The strength can either increase or decrease with a larger probability to decrease. The average ultimate strength is about 4% lower than that of a connection with bolthole position perfectly aligned for any type of connections examined in this study. The effect of bolthole error of the connection on the ultimate strength is larger for the connection with steel cover plates than that with FRP cover plates. The coefficients of variation of the ultimate strength of the two, three, and fourrow bolted connections with steel cover plates are 13.0, 15.6, and 13.9%, respectively, whereas those of connections with FRP cover plates are 3.2, 5.0, and 5.9%, respectively. 
(4) Probability distribution of the ultimate strength due to bolthole position errors for the two to fourrow bolted connections with steel cover plates and tworow bolted connections with FRP cover plates can be modeled by a normal distribution, and that of the three and fourrow bolted connections with FRP cover plates can be modeled by a Weibull distribution with a significance level of 0.05. Ninety five percent nonexceedance strength of the connection with FRP cover plates is about 85% of the connection with bolt positions perfectly aligned, while that of the connection with steel cover plates is about 71% of the perfect bolt position case. 
References 

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