A failure-envelope-based method for the probabilistic failure prediction of composite multi-bolt double-lap joints

Composites Part B 172 (2019) 593–602

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A failure-envelope-based method for the probabilistic failure prediction of composite multi-bolt double-lap joints Fengrui Liu a, b, c, Ziang Fang a, b, Libin Zhao a, b, *, Jianyu Zhang d, **, Ning Hu d a

School of Astronautics, Beihang University, Beijing, 100191, PR China Key Laboratory of Spacecraft Design Optimization and Dynamic Simulation Technologies, Minisitry of Education, Beihang University, Beijing, 100191, PR China c State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an, 710049, PR China d College of Aerospace Engineering, Chongqing University, Chongqing, 400044, PR China b

A R T I C L E I N F O

A B S T R A C T

Keywords: Laminates Strength Statistical properties/methods Mechanical testing Joints/joining Bolted joints Failure envelope method laminates

A prediction method is proposed for the probabilistic failure of composite multi-bolt double-lap joints based on a single-parameter spring-based method, a failure envelope method and Monte Carlo simulation. In this method, the effects of the randomness of the laminate bearing and open-hole strength, laminate’s properties, geometric parameters, clearances, tightening torques and bolt bearing chord stiffness are considered. And the application of failure envelope method leads to an advantage of a very small amount of calculation. To validate the proposed method, composite two-bolt and three-bolt double-lap joints were designed and tested to determine the sto­ chastic failure load. Following the testing of all the necessary parameters, satisfactory agreement between the numerical and experimental stochastic failure loads of the multi-bolt joints was found. Furthermore, the sensi­ tivities of the probabilistic failure load of multi-bolt joints were investigated. It follows that the randomness of the laminate open-hole tensile and bearing strengths have the most significant influences on the failure load, while the influences of the randomness of the other parameters are slight, which affect bolt load distribution directly, and indirectly affect the failure load through influencing the bolt load distribution.

1. Introduction Carbon fibre reinforced resin composite materials [1–5] have been widely used in the aerospace industry for their high strength-to-weight ratios and superior stiffness-to-weight ratios. Bolted joints are impor­ tant joint forms, and their failure is influenced by many factors, such as material mechanical properties, ply angles [6], end distances [7], bolt geometries [8], clearances and bolt tightening torques [9,10]. Mean­ while, because of the remarkable scattering of these factors, substantial uncertainty exists in the failure loads of bolted joints. Thus, the proba­ bilistic strength analysis of composite joints has long been a research hotspot [11–14]. Nakayama et al. [11] applied a method combining FE damage analysis and a Monte Carlo method to pin joints in CFRP lam­ inates to analyse the probabilistic strength of a composite single-bolt joint, where the probabilistic distribution of the bearing strength caused by the randomness of the material and dimension parameters was evaluated. Li [12] proposed a computational framework based on the maximum entropy method and a 2D FE model that takes into

account large deformation theory and the non-linear shear behaviour of the composite material to conduct a probabilistic strength prediction of a single-bolt joint. Li et al. [13] utilized subset simulation and 2D FE analysis to present an methodology to predict the mean value, coeffi­ cient of variation and cumulative distribution function curve of the bearing strength of a composite single-bolt joint, where three groups of failure criteria, three sets of degradation rules and two types of shear relationships were employed for a progressive damage analysis. Zhao et al. [14] presented a probabilistic strength prediction model by combining the modified characteristic curve method and Monte Carlo simulation. Their probabilistic strength study, considering the randomness of the basic design parameters for a typical composite double-lap single-bolt joint, concluded that the longitudinal compres­ sive strength, ply thickness and longitudinal elastic modulus of the unidirectional lamina are the key factors affecting the probabilistic failure load and reliability of the joint. However, the majority of studies have focused on single-bolt joints, while only a few have addressed multi-bolt joints. Askri et al. [15] studied the failure probability of

* Corresponding author. School of Astronautics, Beihang University, Beijing, 100191, PR China. ** Corresponding author. College of Aerospace Engineering, Chongqing University, Chongqing, 400044, PR China. E-mail addresses: [email protected] (L. Zhao), [email protected] (J. Zhang). https://doi.org/10.1016/j.compositesb.2019.05.034 Received 21 September 2018; Received in revised form 24 April 2019; Accepted 5 May 2019 Available online 8 May 2019 1359-8368/© 2019 Elsevier Ltd. All rights reserved.

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Composites Part B 172 (2019) 593–602

multi-bolt joints, but the failure prediction was based on the finite element method, which requires a large amount of calculations and complex computation, particularly considering the many factors involved in the probability analysis. Therefore, research on a method for the probabilistic strength prediction of composite multi-bolt joints that has high calculation efficiency would be valuable for engineering applications. The prediction of the static failure of multi-bolt joints in engineering involves two steps: a bolt load distribution analysis and a critical hole failure analysis [16–18]. The bolt load distribution analysis methods include finite element methods [19], analytical methods [20], and stiffness methods [21]. Finite element methods require a large amount of computation and are therefore not suitable for a probabilistic bolt load distribution analysis. Analytical methods are also not preferred for this type of analysis because only a few factors can be considered, leaving many significant factors overlooked, particularly the tightening torques. Stiffness methods have thus seen relatively frequent usage in the study of bolt load distribution probabilities [22–24]. For example, after Liu et al. [21] presented an analytical tri-linear joint stiffness model accounting for the effect of bolt-hole clearances on the bearing chord stiffness and an improved stiffness method, Liu et al. [24] utilized the improved stiffness method and Monte Carlo simulation to analyse the sensitivity of the influence of the clearances and tightening torques on the bolt load distribution randomness. Failure analysis methods for the critical hole include progressive damage methods [25], characteristic curve methods [16,17] and failure envelope methods [18,26]. Progressive damage methods require bolt-hole finite element models. The contact nonlinearities and material nonlinearities are also needed for accurate stress and material failure process predictions, respectively. Progressive damage methods can be used to obtain failure loads and damage processes. Li [12] and Li et al. [13] used probability analysis methods and progressive damage methods to study the strength deviation of single-bolt joints. However, the large number of calculations makes these methods unsuitable for engineering applications of probabilistic failure analysis, and the ob­ tained damage process, which necessitates an extreme amount of computation, is not feasible for the probability analysis. There is also an additional defect that the failure criteria and material degradation rules, etc., need additional applicability analyses for their usage in structures. Zhang et al. [25], for instance, identified a suitable progressive damage model for bolted joints by comparing two types of strength parameters, two groups of failure criteria and two sets of degradation rules. Li et al. [13] compared three groups of failure criteria, three sets of degradation rules and two type of shear relationship in failure probability analyses of bolted joints. These applicability analyses, though, are quite laborious. Characteristic curve methods also require bolt-hole finite element models, and contact nonlinearities must again be considered, to accu­ rately determine stresses [16,17]. Because the material nonlinearities do not need to be considered, the calculation cost is smaller than that of the progressive damage methods. With the research basis of the proposed three-parameter characteristic curve by introducing a shear-out char­ acteristic length to the existing characteristic curves [16] and proposed suggestions for testing methods of the characteristic length [17], a probabilistic strength analysis method was proposed using characteristic curve methods for failure prediction [14]. However, the characteristic curve methods suffer from a series of problems: the influence of the joint geometric dimensions on the characteristic lengths is great, but the in­ fluence rule is hard to acquire, and the finite element models require a large amount of calculation. In contrast, in the failure envelope methods, after obtaining the necessary parameters, the strengths of the critical holes with different geometric sizes can be predicted directly with relatively little calculation [18,26]. Hart-Smith [27] originally proposed the failure envelope method, and Liu et al. [18] modified this method by considering the influence of the bypass load on the bearing strength, which significantly improved the prediction accuracy for the failure load of two-bolt and four-bolt joints. Zhao et al. [26] also

conducted experimental and numerical comparative studies and sug­ gested a calculation method for the stress concentration relief factors, which are critical parameters for calculating the slopes of the failure envelope. The failure envelope method requires only a small amount of calculation, making it the most suitable method for probability distri­ bution research. Various methods have been used to estimate the randomness. Li [12] applied the maximum entropy method. Li et al. [13] utilized subset simulation. Askri et al. [15] applied a Sobol sequence based quasi-Monte Carlo sampling method. Choi and Grandhi [28] and Hamdia et al. [29] used polynomial chaos sampling to represent the response of an un­ certain system. The above methods are all simplified methods that predict the probability distributions of the analysis targets with fewer samples, so the acquired probability distributions are only approximate results. Monte Carlo simulation, on the other hand, is a non-simplified method that samples indiscriminately from the entire random param­ eter feasible region, and the outcomes are considered to be exact solu­ tions. Some researchers have validated the above simplified methods with Monte Carlo simulation [15]. Davidson and Waas [30], Kumar and Bhat [31], and Zhao et al. [14] utilized Monte Carlo simulation for analysing structural failure probabilities. One shortcoming of the Monte Carlo simulation, though, is the need for massive sampling caused by the indiscriminate sampling, which will cost a lot of time for each proba­ bilistic failure analysis. According to the above analyses, the application of the stiffness method and the failure envelope method can reduce the amount of calculation required in each failure analysis. Therefore, considering the lack of an exact solution in probability characteristic studies of the failure of multi-bolt joints, the Monte Carlo simulation was chosen in this paper to achieve an accurate and convincing result. A probability analysis method to address the failure of composite multi-bolt double-lap joints combining the spring-based stiffness method, the failure envelope method and Monte Carlo simulation is proposed in this paper. The influences of the material’s elastic proper­ ties, geometric sizes, clearances, tightening torques, bolt bearing chord stiffness, bolt bearing strength and open-hole tensile strength are considered. To verify the proposed method, two-bolt and three-bolt joints are designed and tested to obtain the failure loads and their probabilistic characteristics. Following the design and testing of singlebolt double-lap joints and open-hole tensile laminates to provide the basic parameters and their probabilistic characteristic, the proposed method was validated by comparing the prediction results with exper­ imental results. Finally, the influence sensitivities of the various multibolt joint parameters are studied. 2. Analysis method for the random failure of multi-bolt doublelap joint The stiffness method and the failure envelope method are intro­ duced, and as an extension of the two methods, how to consider random variables in the two methods is described in detail. The selection scheme and the transfer process of random variables are summarized, and the detailed process using Monte Carlo simulation to predict probabilistic failure is explicated. 2.1. Stiffness method considering random parameter distributions Fasteners and connecting laminates of double-lap joints can be idealized as one-dimensional elastic elements along the loading direc­ tion according to the stiffness method [24], as shown in Fig. 1, where Fi, ki, and δi (i ¼ 1,2, …,n) indicate the bolt load, bolt stiffness and bolt longitudinal deformation (along the loading direction) corresponding to O the ith bolt, respectively. FM i and F i indicate the inner forces of the laminates between the ith and (iþ1)th bolt, in which the subscripts “M” and “O” indicate the middle laminate and outer laminates, respectively. O kM i and ki are the equivalent stiffnesses of the middle laminate and outer

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Fig. 1. Spring-based stiffness model of multi-bolt double-lap joints.

laminates, respectively. ki indicates the relationship between Fi and δi, which is not a simple proportion function but a relatively complex function and is therefore described as Fi ¼ f(δi). According to Fig. 1, the single-parameter recursive formulae [21] for bolt load predictions are expressed as: 8 < F1 ¼ f ðδ1 Þ FM ¼ F F1 (1) : 1 O F 1 ¼ F1 8 � δ ¼ δi þ F Oi kOi > > < iþ1 Fiþ1 ¼ f ðδiþ1 Þ ¼ FM Fiþ1 > FM i > : iþ1 FOiþ1 ¼ FOi þ Fiþ1 �

δn ¼ δn 1 þ FOn Fn ¼ f ðδn Þ

1

� O kn

where EL is the longitudinal homogeneous Young’s modulus of the connecting laminate, W is the width of the laminates, hM and hO are the thicknesses of the middle laminate and the sum of the thicknesses of the two outer laminates, respectively, and Li is the bolt pitch. EL itself is random, but its values for the middle laminate and outer laminates are taken to be identical because they are typically obtained from the same production batch. The multi-bolt joints are typically multi-column joints, and W is calculated by the laminate width divided by the col­ umn number; thus, it is reasonable to use the same W for the middle laminate and outer laminates. h is the product of the lamina thickness t and the number of layers. The lamina thicknesses t of the middle lami­ nate and the outer laminates are also identical because they come from the same production batch, so a linear relationship between hM and hO is considered. Because the middle laminate and two outer laminates are fixed together before drilling, L itself is random, but the values of the bolt pitches for the two types of laminates are the same. EL is calculated using classical laminate theory corresponding to the reciprocal of the first element in the flexibility matrix of the laminates EL ¼ 1/a11. The laminate stiffness matrix [A] ([A] ¼ [a] 1) is the accumulation of the stiffness matrixes for each lamina:

� M ki FM i (2)

1

� M FM n 1 kn 1

(3)

Define the sum of the bolt loads as Fsum: n X

Fi

Fsum ¼

(4)

i¼1

1 A*ij ¼ P n

There is only one variable δ1 in Eqs. (1)–(3) for calculating bolt loads; that is, the bolt loads Fi can be calculated according to Eqs. (1)–(3) after assuming a value of δ1. Fsum can then be calculated according to Eq. (4). That is, there is a one-to-one functional relationship Fsum ¼ f(δ1). In addition, the Fsum ¼ f(δ1) was proven to be a monotonically nondecreasing function [21]. Thus, the bolt load distributions can be solved with a one-dimensional search method: reduce or increase δ1 when the sum of all bolt loads Fsum is larger or smaller than the external load F, respectively, until the difference between Fsum and F is smaller than a given tolerance. The detailed solution process has been provided in the literature [21] and will not be explored in this paper. The toler­ ance of the difference between Fsum and F was set to be 1 � 10 6 in this paper for the precise control of the one-dimensional search, which sat­ isfies the precision requirements for the failure load predictions of multi-bolt joints. After obtaining the bolt load Fi of the joints, the bolt bearing/bypass load ratio Ri (or “bolt load ratio” for short) of bolt hole i of the middle laminate can be calculated with Fi: Fi Ri ¼ P n

n X

(7)

ðmÞ

Qij tm tm m¼1

m¼1

where Qij

ðmÞ

represents the lamina off-axis stiffness coefficient of layer m.

The off-axis stiffness matrix Q can be calculated from the lamina on-axis stiffness matrix Q and the ply orientation angle conversion matrix T: (8)

Q ¼ T 1 QTT 2 where 2

T ¼

4

cos2 θ sin2 θ sin θ cos θ

sin2 θ cos2 θ sin θ cos θ

3

2 sin θ cos θ 2 sin θ cos θ 5, cos2 θ sin2 θ

Q ¼

3 Q11 Q12 0 E2 4 Q12 Q22 0 5, Q11 ¼ E1 v12 E2 , Q12 ¼ E1E2vv212E2 , Q22 ¼ E1 E2vE21 E2 , and 12 12 12 0 0 Q66 Q66 ¼ G12 . θ is the ply orientation angle, and superscripts “-1” and “T” represent the inverse and transpose of a matrix, respectively. The bolt stiffness model f(δ) strongly affects the bolt load ratio pre­ dictions and has been developed from the proportional model [32] to a two-stage model that takes the clearances into account [33]. A three-stage model considering both the tightening torques and clear­ ances [34] is also proposed and is indicated with the polygonal lines composed of thick green, red, and blue lines in Fig. 2. The three-stage model is expressed as: 8 1 � 0 � �δ < Fc k1 �
(5) Fj

j¼iþ1

where Fi is the bolt load at hole i of the middle laminate corresponding to P bolt i (bolt hole i of the middle laminate for short) and nj¼iþ1 Fj is the O bypass load of bolt hole i of the middle laminate. kM i and ki in Eqs. (2) and (3) are expressed as: � kji ¼ EL Whj Li ðj ¼ M; OÞ (6)

where k1 is the joint stiffness for the initial quasi-linear region (Region I 595

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Fig. 2. Bolt stiffness model considering parameter randomness.

in Fig. 2); Fc is the critical friction indicating the height of the transition region (Region II in Fig. 2); c is the bolt-hole clearance, which is the length of the transition region; and k3 is the bolt bearing chord stiffness corresponding to the stiffness of the bolt load transmission region (Re­ gion III in Fig. 2). Liu et al. [21] and Mccarthy and Gray [34] treated k1 as a constant. Fc is a proportional function of the tightening torque T (Fc ¼ ηT), where η is a coefficient. Liu et al. [21] treated k3 as a linear function of the clear­ ance. Because of the large scatter of the fitting results between k3 and c in Ref. [21], k3 is assumed to be random and independent of c in this paper to obtain conservative results. Thus, the random variables of the bolt stiffness model are c, T and k3. The � 3σ scatter band of the bolt stiffness is shown in Fig. 2 with dotted and dashed lines. Overall, the random variables that affect the randomness of the bolt load ratios include material performance parameters (E1, E2, ν12, G12, θ, and t), geometric dimensions (W, hj, and Li) and random parameters of

Fig. 3. Failure envelope strength parameters.

considering

the

randomness

of

the

is also shown in Fig. 3. Notably, Fbear and Fby are affected by the randomness of the material properties, layer angles, single-layer thick­ nesses, clearances and tightening torques, so the effects of these pa­ rameters on the strength randomness have been considered. 2.3. Randomness transmission relationships from the variables to the predicted failure loads All of the random variables of multi-bolt joints considered in this method are listed in the 4th column from the left in Table 1. The transmission processes of the randomness from the variables to the predicted failure load are shown on the right of Table 1. The randomness of variables E11, E22, G12, ν12, θ1, θ2, θ3 and θ4 induces randomness in EL; the randomness of t induces randomness in hj; the randomness of vari­

the bolt stiffness model (Ti, ci, and k3i ).

ables EL, hj, W and Li induces randomness in ki ; the randomness of j

2.2. Failure envelope method considering random parameter distributions

variables ci, Ti and j

Hart-Smith [27] defined the stress concentration factor of a com­ posite laminate with a hole as the ratio of the laminate’s strength (without a hole) to the average failure stress of the laminate with a hole. It is further assumed that a proportional relationship exists between the stress concentration factors of composite laminates and isotropic ma­ terials. The proportionality coefficient is referred to as the stress con­ centration relief factor of the composite laminate. Hart-Smith also noted that only a bearing failure or tensile failure can occur when the joints are reasonably designed. Therefore, a hole strength analysis method that evaluates the bolt bearing/bypass load ratio for multi-bolt double-lap joints, referred to as the failure envelope method, is proposed. Liu et al. [18] modified this method and improved the prediction accuracy for the failure loads of two- and four-bolt joints. This modified failure envelope is expressed with the black solid lines marked A, B, and C in Fig. 3. In Fig. 3, point A represents the bearing failure load Fbear; point C represents the open-hole tensile failure load Fby; ηAB represents the slope of line AB and is obtained by combining the experimental results with the finite element results [18]; and ηBC represents the slope of line BC and is equal to the negative ratio between the open-hole tensile stress concentration factor and the load-hole tensile stress concentration factor [26]. The randomness of Fbear and Fby is experimentally determined. ηAB and ηBC are deduced from the average results of experiments, including the abovementioned experiments. Therefore, their randomness is neglected to avoid the repeated consideration of the randomness. Thus, the random variables of the failure envelope are Fbear and Fby, which are tested with single-bolt joints and open-hole tensile laminates, respec­ tively. For the failure envelope with randomness, the � 3σ scatter band

k3i

induces randomness in f(δi); the randomness of

variables ki and f(δi) induces randomness in Ri; the randomness of var­ iables Fbear and Fby induces randomness in the failure envelope; and the randomness of Ri and the failure envelope induces randomness in the predicted failure load of the multi-bolt joints. The right arrow ‘→’ in­ dicates a causal relationship. All variables are divided into 10 categories for the subsequent analysis. A truncated normal distribution function N (μ,σ ) [24] is adopted in modelling the randomness of the 10 categories of variables to ensure that the sampling values of the random parameters are located within the allowable tolerance band: � x μ 1 σϕ σ f ðx; μ; σ Þ ¼ (10) Φð þ 3Þ Φð 3Þ where ϕ and Φ are the probability density function (PDF) and cumula­ tive distribution function (CDF) of the standard normal distribution, respectively. In the prediction of the random failure load of multi-bolt joints with the proposed failure analysis strategy, a set of values for the 10 cate­ gories of variables are first sampled according to their truncated normal distribution functions. Then, the failure envelope can be obtained with 0 0 the sampled F bear and F by , which is the polygonal line ABC in Fig. 4. The bolt load ratio for each hole in the middle laminate corresponding to 0 Bolt i R i , which changed over the external load, was calculated with the stiffness method and is drawn in Fig. 4 with the blue curve (the abscissa of the bolt load ratio curve is the bypass load and the ordinate of the bolt load ratio curve is the bolt load, and thus, the slope of the curve is the bolt load ratio). The intersection point of the bolt load ratio curve and 596

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Table 1 Random variables and the transmission processes of randomness. No.

Category no.

Name

Random variable

Transmission process of randomness

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 (Material properties)

E11 E22 G12 v12 θ1 θ2 θ3 θ4 T W Li (i ¼ 1,2, …n-1) ci (i ¼ 1,2, …n) Ti (i ¼ 1,2, …n)

→EL

3 4 5 6 7 8

Young’s modulus Young’s modulus Shear modulus Poisson’s ratio in plane 0� 90� 45� 45� Thickness of lamina Width of laminates Bolt pitches Bolt-hole clearances Bolt tightening torques Bolt bearing stiffness

15 16

9 10

Bearing strength for loaded hole Open-hole tensile strength

Fbear Fby

2 (ply angles)

k3i (i ¼ 1,2, …n)

→hj → → → → → → →

j

→ki

→Ri

→Failure loads of multi-bolt joints

→f(δi)

→ →

→Failure envelope

3. Validation 3.1. Multi-bolt double-lap joints To validate the proposed method, two types of multi-bolt double-lap joints (the two-bolt joint and the three-bolt joint) were designed and manufactured, with ten specimens manufactured for each joint. The static strength experiments were conducted as per ASTM 5961 [36], and the random characteristics of the failure loads were acquired. The con­ figurations of the two types of joints are shown in Fig. 5, where the values of the dimensions without randomness are indicated, and the definitions of the random variables are marked. The composite laminates are made of T800/X850 [7,8] carbon/­ epoxy prepreg containing the medium-modulus high-strength carbon fibre tow 12 K from the Japanese Mitsubishi®, where X850 is a high-toughness epoxy resin solidified at elevated temperature. The stiffness of the lamina is listed in Table 2, and the strength parameters are as follows: XT ¼ 3071 MPa, XC ¼ 1747 MPa, YT ¼ 88 MPa, YC ¼ 271 MPa, and S12 ¼ 143 MPa. The stacking sequence of the lami­ nates is [45/0/-45/0/90/0/45/0/-45/0]s. The fasteners are HST12 high-locking bolts [37] with matching HST1078 nuts [38]. The mean values μ and standard deviations σ of the random variables are listed in Table 2. μ and σ for the material parameters E11, E22, G12 and ν12 in Table 2 are calculated according to 15 data points acquired from the material producer. The mean values of the ply orientation angles θ1-θ4 are 0� , 90� , 45� and 45� , and the standard deviations are all set to 0.9 [24]. The μ and σ of the lamina thickness are 0.185 mm and 0.003 mm, respectively [24]. The mean values of the laminate width W and bolt pitches Li are 30 mm and 20 mm, respectively, and the standard deviations are calculated to be 0.33 mm according to the tolerance band of �1 mm in ASTM 5961 [36] and Eq. (11): � μ ¼ ðxmax þ xmin Þ=2 (11) σ ¼ ðxmax xmin Þ=6

Fig. 4. Failure prediction of multi-bolt joint in one sampling.

the failure envelope denotes that the failure of the hole will occur, so the bolt load ratio of the intersection point is the bolt load ratio for the failure of this hole i, and the external load corresponding to the failure bolt load ratio of this hole is the failure load Fifailure predicted by this hole. The predicted failure load of the joint F failure in this sampling is the 0 minimum of all Fifailure , that is F failure ¼ minðFifailure Þ. Then, the proba­ 0

bility characteristics for the random failure loads F failure are calculated after further sampling. The sampling and calculations are conducted until the probability characteristics become stable. Note that the mean values and the standard deviations of some variables in Fig. 1 are time-dependent random variables, such as the material properties varying with the storage time in hygrothermal conditions [35]; the tightening torque relaxation over time due to the viscoelastic response of the composite laminate or slip in the bolt threads [9,10]; and Fbear and Fby being affected by the randomness of the ma­ terial properties and the tightening torques and being time-dependent random variables. Because of lacking in the variations of these random parameters with time, their time-dependent mean values and the standard deviations are not used in this model. However, the strength probability of the joints at a certain moment could be predicted with this method considering the adopted mean values and standard deviations of the material properties, the tightening torques, and Fbear and Fby at that particular moment. 0

where xmax and xmin are the allowable maximum and minimum of the parameters, respectively. For example, the maximum and minimum of W are 31 mm and 29 mm, respectively. The minimum and maximum clearances are set to be 0 and one percent of the hole diameter D (0.0476 mm) [39], respectively, and then μ and σ are calculated as 0.024 and 0.008, respectively. In HST1078, the tightening torque T is specified with a range from 4.52 Nm to 5.65 Nm [38], so μ and σ are calculated with Eq. (11) as 5.09 Nm and 0.19 Nm, respectively. The μ and σ of the two variables, including the bolt bearing chord stiffness and bolt bearing strength, are tested with single-bolt joints, and the μ and σ of the open-hole tensile strength are tested with open-hole tensile laminates. The parameters k1 and η in the bolt stiffness models 597

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Fig. 5. Configurations of specimens: (a) single-column two-bolt joint (b) single-column three-bolt joint.

bearing chord stiffness, bolt bearing strength, and open-hole tensile strength) are tested, and the parameters k1 and η are measured.

Table 2 Mean values and standard deviations of the random variables. No.

Category no.

Name

Random variable

μ

σ

Remark

1

1 (Material properties)

Young’s modulus Young’s modulus Shear modulus Poisson’s ratio in plane 0� 90� 45� 45� Thickness of lamina Width of laminates Bolt pitches

E11 (GPa)

195

3.8

Measured

E22 (GPa)

8.58

0.086

Measured

G12 (GPa)

4.57

0.098

Measured

v12

0.34

0.020

Measured

θ1 (degree) θ2 (degree) θ3 (degree) θ4 (degree) t (mm)

0 90 45 45 0.185

0.9 0.9 0.9 0.9 0.004

[25] [25] [25] [25] [25]

W (mm)

48

0.33

[35]

Li (i ¼ 1,2, …n-1) (mm) ci (i ¼ 1,2, …n) (mm) Ti (i ¼ 1,2, …n) (Nm)

48

0.33

[35]

0.023

0.008

[36]

5.09

0.19

[34]

k3i (i ¼ 1,2, …n) (kN/ mm)

To be measured

Fbear (kN)

To be measured

Fby (kN)

To be measured

2 3 4 5 6 7 8 9

2 (Ply angles)

10

4

11

5

12

6

13

7

14

8

15

9

16

10

3

Bolt-hole clearances Bolt tightening torques Bolt bearing stiffness Bearing strength for loaded hole Open-hole tensile strength

3.2.1. Specimens The configurations of the single-bolt double-lap joint and open-hole tensile laminate are designed as per ASTM 5961 [36], as shown in Fig. 6. For the single-bolt double-lap joint and open-hole tensile laminate, the materials, stack sequences, and fasteners are the same as those for the multi-bolt joints shown in Fig. 5. The experiments were conducted as per ASTM 5961 [36] on an INSTRON-8803 hydraulic pressure servo material testing machine, and an extensometer was used for measuring the hole deformations. 3.2.2. Experimental results of the stiffness model Load vs. hole deformation curves of fifteen single-bolt double-lap joints are shown by the black solid line in Fig. 7. The preload, which is 30% of the predicted maximum failure load and higher than the critical friction in Eq. (8), was loaded and unloaded before testing to obtain a straighter initial quasi-linear region and bolt load transmission region. Similarly, prominent inflexions between the two regions can be found in Fig. 7(b), and Fc can be calculated with high precision. The test data are fitted, and the stiffness model parameters are ac­ quired. The initial stiffness k1 is 202.42 (kN/mm); the average value of Fc is 6.30 kN; η ¼ 6.30/5.09 ¼ 1.24; and μ and σ of k3 are 47.11 (kN/mm) and 3.57 (kN/mm), respectively. Combined with the parameters c and T in Table 2, all parameters of the bolt stiffness model are acquired. 3.2.3. Experimental results of the failure envelope parameters Static failure experiments were conducted on fifteen single-bolt double-lap joints, and the maximum loads of the load vs. hole defor­ mation curves are listed in Table 3. Five specimens were manufactured and tested for the open-hole tensile experiments, and the corresponding five maximum failure loads are listed in Table 3. The corresponding normal distribution parameters of Fbear and Fby are also listed in Table 3.

(Eq. (9)) are tested similarly to the single-bolt joints. The slopes of ηAB and ηBC of the failure envelope are set as 0.074 and 0.406, respectively, according to Ref. [26].

3.3. Validation and analysis of the failure load results Monte Carlo simulations with ten million iterations were conducted for both types of joints to obtain stable statistical characteristics. The solid lines in Fig. 8 denote the CDFs of the failure loads. Ten specimens of both types of joints were manufactured, and the CDFs of the failure loads from the experiments are denoted by red points.

3.2. Parameter measurement In this section, the μ and σ of the three variables, (i.e., the bolt 598

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Fig. 6. Configuration of specimens: (a) single-bolt double-lap joint (b) open-hole tensile laminate.

Fig. 7. Load vs. hole deformation curves of single-bolt joints: (a) complete curves (b) initial quasi-linear region and bolt load transmission region of curves.

values of the predicted and experimental results for both joints are less than 3%, indicating that the prediction results are very accurate. For the standard deviation prediction results, the predicted standard deviations of the both joints are larger than that of the experimental results. The reason for the conservative result is that an independent clearance c and bolt bearing chord stiffness k3 are applied despite the linear relationship found by Liu et al. [21]. Similarly, a subtle relationship appears between the clearances and hole failures [40]; however, their independent re­ lationships are also used in this paper for simplifying the calculation. Thus, a positive error for the standard deviation is expected, although 25.48% is still not large. Therefore, with the proposed method, the mean values of the failure loads can be accurately predicted, while the pre­ diction errors of the standard deviations are less than 26%.

Table 3 Experimental results for the strength parameters. Specimen no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Mean value (kN) Standard deviation (kN)

Failure load Single-bolt joint

Open-hole laminate

19.0 20.3 20.3 20.4 20.7 20.8 20.9 21.1 21.2 21.2 21.5 21.7 21.8 22.1 22.3 21.02 0.82

87.8 88.2 91.8 92.1 94.6 – – – – – – – – – – 90.90 2.56

4. Sensitivities of the statistical characteristics of the failure loads Another important purpose of the failure load probability distribu­ tion analysis is to obtain the parameter sensitivity, which has a higher significance than the deterministic parameter sensitivity because deterministic parameter sensitivity research that disregards the ranges of the parameters cannot be applied directly to engineering. For instance, suppose that parameter A has a large deterministic sensitivity but a very small random range, while parameter B has a smaller

As shown in Fig. 8, the test results are all close to the prediction curves. The mean values and standard deviations of the tested and predicted failure loads are listed in Table 4. The errors of the mean 599

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Composites Part B 172 (2019) 593–602

Fig. 8. Comparisons of predicted and experimental CDFs: (a) two-bolt joints (b) three-bolt joints.

approximately 0.56 times that corresponding to “complex”; the value of ΔF corresponding to c1 is approximately 0.17 times that corresponding to “complex”; and the value of ΔF corresponding to T1 is 0.1 times that corresponding to “complex”. These results show that Fbear has a signif­ icant contribution to the deviation of the failure load and that k3, c, and T also have detectable influences. Fig. 9(b) shows that the median values corresponding to all variables are identical and Fby has the greatest contribution to the deviation of the failure loads in three-bolt joints. The CDF corresponding to Fbear is the closest to the median value. From Fig. 9(d), the value of ΔF corre­ sponding to Fby is approximately 0.94 times that corresponding to

Table 4 Errors of the mean values and standard deviations for the prediction results. Joints

Experimental results

Prediction errors (%)

μ

σ

(kN)

Cv (%)

μ

σ

(kN)

Cv (%)

μ

σ

41.39

1.55

3.74

42.14

1.95

4.62

1.81

25.48

60.33

1.68

2.78

59.62

2.07

3.47

(kN) Twobolt joints Threebolt joints

Prediction results

(kN)

1.17

23.41

“complex”; the value of ΔF corresponding to k31 is approximately 0.29 times that corresponding to “complex”, which is less than that in the two-bolt joints; and the values of ΔF corresponding to c1 and T1 are 0.1 times that corresponding to “complex”. Therefore, the main factor affecting the failure load deviation is the open-hole tensile strength Fby, followed by the bolt bearing chord stiffness k3, c1, and T1. The influences of the other parameters are small. The sensitivities of k3, c1, and T1 for the three-bolt joints are less than those for the two-bolt joints. One notable phenomenon appears in the results for the two-bolt joints. The median values of the failure load corresponding to “com­

Here, Cv is the variation coefficient, defined as standard deviation/mean value � 100%.

deterministic sensitivity but a far larger random range. There is a pos­ sibility that parameter B is more important than parameter A. Therefore, the sensitivities of the statistical characteristics of the failure loads were investigated. The randomness of the failure load of multi-bolt joints caused by the randomness of a specific parameter can be calculated with a Monte Carlo simulation when this parameter is randomly sampled and other pa­ rameters are set at their mean values. Monte Carlo simulations of approximately 100,000 iterations were executed to obtain stable dis­ tributions of the failure loads arising from the randomness in each parameter. The CDFs of the failure loads of two-bolt joints and three-bolt joints corresponding to the 10 categories of parameters are shown in Fig. 9(a) and (b), respectively. For each category, the CDF corresponding to the parameter that caused the maximum deviation of the failure load is provided. A CDF marked “complex” denotes a CDF of the failure load when all 10 categories of the parameters are random. The ordinates of the intersection points between the auxiliary lines and the CDF curves are expressed as the failure loads for a 99.5% safe probability F995 and the median value of the failure loads F5. As shown in Fig. 9(a), for two-bolt joints, the CDFs corresponding to the random variables E11, θ1, t, W, L1, and Fby nearly coincide and are very close to the median values of the failure loads. The CDF corre­ sponding to the “complex” designation is the farthest from the median values. The CDF corresponding to Fbear is the farthest from the median

plex” and k31 are different from those corresponding to the other pa­ rameters, which is not the case for the three-bolt joints. This finding

occurs because the randomness of k31 leads to a large deviation in the load distributions. When predicting the failure loads of two-bolt joints with the failure envelope method, the deviation of the load distributions causes a circumstance where the collapsed hole is no longer always hole 1; the resulting collapse probabilities of hole 1 and hole 2 are 66% and 34%, respectively. This condition causes F5 to differ from that in the three-bolt joints. 5. Concluding remarks A prediction model for the probabilistic failure of a composite double-lap multi-bolt joint was developed based on the single-parameter spring-based method, failure envelope method and Monte Carlo simu­ lation. In the proposed method, the basic parameters, including lami­ nate properties, geometric parameters, bolt-hole clearances, tightening torques, bolt bearing stiffnesses, laminate bearing and open-hole tensile strength, were systematically gathered as the random input variables. The failure loads of numerous composite two-bolt and three-bolt doublelap joints were tested, and satisfactory agreement was found between the numerical and experimental stochastic failure loads to validate the proposed method. Moreover, the sensitivities of the failure loads were investigated by using the proposed method. It was concluded that the main factors affecting the failure load deviation are the bolt bearing

when only one parameter is random, and the CDFs corresponding to k31 , c1, and T1 follow. The CDF corresponding to Fbear is the closest to that corresponding to the “complex” designation. ΔF is defined as the dif­ ference between F995 and F5 and is shown in Fig. 9(c). The values of ΔF corresponding to E11, θ1, t, W, L1, and Fby are almost zero. The value of ΔF corresponding to Fbear is approximately 0.83 times that corre­ sponding to “complex”; the value of ΔF corresponding to k31 is

600

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Composites Part B 172 (2019) 593–602

Fig. 9. Sensitivity study of failure loads: (a) CDF for the two-bolt joints (b) CDF for the three-bolt joints (c) F995 and F5 for the two-bolt joints (d) F995 and F5 for the three-bolt joints.

strength and open-hole tensile strength, followed by the bolt bearing chord stiffness, clearance and tightening torque. The influences of the other parameters are small. The sensitivity to the bolt bearing chord stiffness, clearance and tightening torque for the two-bolt joints is higher than that for the three-bolt joints. The variations in the load distribution cause an alternative median value in two-bolt joints but not in three-bolt joints. The probabilistic failure of the composite multi-bolt joint was pre­ liminary studied by now. Two aspects also need to be further researched: 1) The probabilistic failure prediction method integrating the efficient failure prediction method and the time-saving randomness estimating method is needed; 2) The probabilistic experiments with more speci­ mens are conducted to achieve the stochastic failure load distribution, which supports more accurate validations of the probabilistic failure prediction method and sensitivity analysis. The probabilistic failure prediction results and sensitivity analysis results have wide application prospects for reliability analyses and structural designs of large scale composite structures.

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