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Design-oriented strength of mechanical joints in composite laminate structures and reliability-based design factor
Composite Structures 132 (2015) 1–11
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Design-oriented strength of mechanical joints in composite laminate structures and reliability-based design factor Masahiro Nakayama a,⇑, Nobuhide Uda a,1, Kousei Ono a,1, Shin-ichi Takeda b, Tetsuya Morimoto b a b
Department of Aeronautics and Astronautics, Kyushu University, 744 Moto-oka, Nishi-ku, Fukuoka 819-0395, Japan Japan Aerospace Exploration Agency, 6-13-1 Osawa, Mitaka-shi, Tokyo 181-0015, Japan
a r t i c l e
i n f o
Article history: Available online 2 May 2015 Keywords: Composite material Mechanical joint Design Strength Bearing test
a b s t r a c t Mechanical joints may act as bottlenecks in the design process of composite structures. Fitting factors need to be multiplied with the applied load condition to assure reliability. For more efficient design, this paper proposes a brand-new design-oriented strength: knee point strength for the mechanical joints in composite laminates. The knee point strength is derived from the stiffness change rate for the bearing stress–strain behavior in a bearing test. The strength can be correlated with the internal bearing damage; this is verified by micrography about the bearing test specimen and finite element analysis. The probabilistic design and analysis method was applied to acquire the reliability-based fitting factor. An approach combining finite element damage analysis and a stochastic technique was adopted to analyze the probabilistic properties for the design-oriented strength of the bolted joints in CFRP laminates. The probabilistic and deterministic designs were compared and discussed. Then a proper value of the fitting factor was proposed. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Applications of advanced composite materials such as carbon-fiber-reinforced plastics (CFRPs) are expanding in not only the aerospace industry but also other industries because of their high specific strength and stiffness, good fatigue resistance, etc. However, composite materials generally have more scattering properties than conventional metallic materials. Therefore, the ‘‘design tolerance,’’ which is the material strength used in structural design processes, is reduced. The tolerances for composite materials need to be set low because of the large variation in their strength properties. In addition, there is much less experience with using composites as structural materials than with metals. Therefore, structural designs that use composite materials tend to be conservative. One example of conservative structural elements is mechanical joints. Composite-made joints have often been the subjects of many studies both experimentally and analytically [1–9]. In actual applications, mechanical joints are commonly used to assemble structural components; they are necessary because they allow re-access to internal structures and easy replacement of damaged
⇑ Corresponding author. Tel./fax: +81 92 802 3036. 1
E-mail address: [email protected] (M. Nakayama). Tel./fax: +81 92 802 3036.
http://dx.doi.org/10.1016/j.compstruct.2015.04.044 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.
components. They can be critical to strength assessment of the structure because of the stress concentrations their opening holes induce. ASTM standard D 5961 [10] defines the bearing strengths under high loads, which corresponds to the ultimate load state. The testing standard does not define the strengths under relatively low applied loads, which correspond to the limit load state. The ‘‘fitting factor’’ is a design factor with regard to airworthiness [11]; it is multiplied with external loads separately from the safety factor. The fitting factor is greater than or equal to 1.15 [11]. However, the value for joints in CFRP laminate structures should be higher than 1.15 because of the previously noted uncertainty and lack of empirical knowledge. Therefore, more reasonable definitions for the design strength and a suitable design factor are desirable for efficient design of mechanical joints in CFRP structures. This paper proposes a brand-new design-oriented strength for mechanical joints in composite laminates and a reasonable design factor for the joints. Two approaches are adopted. Knee point strength FKP is a design-oriented strength based on an experimental approach. The strength is acquired by focusing on stiffness reduction during bearing behavior, which is unfavorable for actual structures. Former researches about the bearing strength [2,12,13] seem that they focus on nonlinearity of bearing load-displacement relationship. Our research related the nonlinearity, especially in the early steps in the bearing curve, to internal damages in the
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bearing-loaded specimens. We developed a method to identify the starting point of the stiffness reduction, which would be used as a criterion for the limit stress, defined as the stress generated in the composite joint when a limit load is applied to the structure [14]. This study utilized micrography and damage propagation analysis by finite element method to reveal the relationship between the internal bearing damage and the stiffness reduction. The second analytical approach sets the fitting factor based on the reliability of the joint. To obtain greater efficiency in the design of structures made from composites, attempts have been made to introduce a probabilistic or reliability-based design method [15– 17]; this is generally called the probabilistic design and analysis (PDA) method. This method optimizes structures on the basis of their probability of failure or reliability; it should be able to avoid too conservative designs and more fully utilize the good mechanical properties of composite materials. Damage propagation analysis using the finite element model to calculate FKP was combined with the Monte Carlo method to estimate the stochastic properties
of the design strength. Allowable stresses by probabilistic and deterministic designs were compared, and the probabilistic one was translated into the form of fitting factor. The PDA method can be used for providing reasonable fitting factors for composite structures. 2. Experimental 2.1. Bearing test The bearing test is a standard method for evaluating the properties of composite joints. Fig. 1 shows a simplified image of the test and a photograph of an actual system. The CFRP specimen had a fastener hole, and a bearing load was applied through the fastener. Extensometers were equipped on both edges of the specimen to measure relative displacements between the specimen and testing jig. In this study, Toray T800S/3900-2B [18,19] prepreg was used to fabricate the laminates. The composite laminate
Fig. 1. Schematic and actual photograph for the bearing test. (a) Schematic for bearing test. (b) Actual photograph.
M. Nakayama et al. / Composite Structures 132 (2015) 1–11
3
stacking sequence was [45/0/45/90]2S (quasi-isotropic); therefore, the thickness of the laminate was about 3.0 mm. Failure modes of the bearing test specimens [10] vary depending on stacking sequence of the specimens, dimensions of the specimens, and so on. The stacking sequence of our specimens was decided to cause failures in bearing mode (not shear-out or net tension modes). Fig. 2 shows the dimensions of the bearing test specimen. The solid curve in Fig. 3 is a typical bearing stress–strain relationship for the bearing behavior. The zigzag curve in the upper zone of the figure represents the stiffness change rate (SCR) for the bearing behavior; the SCR is explained in the next section. The bearing stress rb and strain eb are defined as follows [10]:
rb ¼ P=ðD hÞ eb ¼
d1 þ d2 2D
ð1Þ ð2Þ Fig. 3. Typical bearing stress–strain diagram.
where P is a load that is applied to the test system, D is the diameter of the fastener hole in the laminate, h is the thickness of the laminate, and di (i = 1, 2) are displacements measured by each of the two extensometers. The stress rb rapidly rose at first because of static friction between the testing jigs and specimen fastened to them. The bearing load increased to a certain degree; the specimen then slipped in the jigs, and the specimen and fastener bolt came into contact. The contact area then bore most of the load. The bearing stress increased linearly initially; after a certain stress level (i.e., defined as the ‘‘knee point’’ in this paper), the curve showed nonlinear behavior. During the nonlinear behavior, internal damage proceeded; the curve finally reached the maximum bearing stress, which is called the ultimate bearing strength Fb in the testing standard [10]. The term ‘‘knee point’’ suggests the starting point of the nonlinear bearing behavior. 2.2. Stiffness change rate This study focused on the change in stiffness of the bearing behavior. The stiffness decreased gradually during the bearing load process. This reduction can be correlated with the internal damage. The zigzag curve in Fig. 3 shows a second-order differential of the rb–eb curve, which corresponds to the rate of change in the bearing stiffness and is hereafter called the SCR:
SCR ¼
@ 2 rb @ e2b
ð3Þ
The SCR was plotted based on sampled data of the bearing test. The data were acquired in discrete form; therefore, the differential should be converted into the central difference. The absolute values of the SCR seem to be irrelevant. Thus, the vertical axis was not
included in the diagram. However, the scales for all SCRs (include later ones) should be standardized. The differences were strongly affected by the sampling rate of the data logger and signal noise. The sampling rate was 2.0 Hz, and the data points were thinned to have an interval of 0.2% bearing strain to remove the effect of the noise. These values were determined empirically. Then second central difference calculations were performed based on the thinned data points and the result was plotted in Fig. 3 as the upper zigzag curve. The curve had some local minima, which suggests that the bearing stress–strain curve had some ‘‘knees.’’ These were assumed to be induced by damage to the laminate. Fig. 4 shows the micrographs of the cross section of the specimens with different states of damage. These specimens are different from the one used to draw the curve in Fig. 3, but all tests were conducted under similar conditions. The specimens shown in Fig. 4 were loaded until their bearing strength reached 270, 380, 490, and 590 MPa. Fig. 5 shows the rb–eb curves and the SCRs of the specimens. However, the 270 and 380 MPa specimens do not have SCRs because of their insufficient data points. The SCRs of the other more highly loaded specimens suggest that the 270 and 380 MPa specimens had positive or slightly negative SCR values. For the 270 MPa bearing stress, where a local minimum did not appear on the SCR curve, no damage was observed in 0° plies, as shown in Fig. 4 (the chip on the left-hand side was made when the specimen was sanded for microscope observation). At 380 MPa, the outer 0° layers had compressive failures near the bearing-loaded areas. The failures were observed to propagate with the bearing load (see photographs for 490 and 590 MPa). Therefore, stiffness reduction is involved with the compressive damages in 0° layers; this result was supported by FE analysis and is later discussed in Section 3.
D = 6.00 + 0.03 – 0.00 mm W = 36 mm L = 135 mm h = 3.0 mm e = 18.0 mm
Fig. 2. Dimensions of bearing test specimen.
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-45
0 45 90 -45 0
45 90 90 45 0 -45 90 45 0 -45
270 MPa
-45 0 45 90 -45
0 45 90 90 45 0 -45 90 45 0 -45
380 MPa
-45 0 45 90 -45
0 45 90 90 45 0 -45 90 45 0 -45
-45 0 45 90 -45
0 45 90 90 45 0 -45 90 45 0 -45
490 MPa
590 MPa
Fig. 4. Micrographs of cross sections on the bearing load line in specimens.
2.3. Knee point on the bearing stress–strain curve The testing standard ASTM D5961 [10] defines two types of bearing strength: (1) the maximum value of the bearing stress Fb and (2) the stress at an intersection of the rb–eb curve, and a line whose slope is equal to the bearing stiffness Eb given a 2.0% offset strain. The strength is called the 2.0% offset strength F2.0. In addition, some researchers have proposed more physical-based definitions of the bearing strength [2,4,13]. However it seemed that they used the nonlinearity as just the threshold for the bearing strength determination. The nonlinearity was not related to the internal bearing damages. For an actual structure, a reduction in stiffness before an applied load reaches its limit value is undesirable. Thus, the knee point strength (hereafter referred to as FKP), which is the stress of the first local minimum of the second-order differential of the rb–eb curve, is proposed as the design strength for bolted joints in composite laminates. FKP is simple to detect, and there is no need for further experimental equipment. It provides a more physical-based design strength value than conventional alternatives. However, there are issues regarding the accuracy in determining the knee point on the rb–eb curve. We developed a method that uses the fourth-order polynomial approximation of the rb–eb curve. First, based on the SCR curve calculated from thinned data points, a temporary knee point is determined. Next, a region near the temporary knee point on the rb–eb curve is approximated by a fourth-order polynomial. The range of ±30 MPa around the temporary knee point was empirically found to be suitable for the
approximated region. The second-order differential of the polynomial suggests the SCR of the approximated region, and the curve of the second-order differential must have a local minimum. The strain value at the local minimum is the knee point strain eKP. By substituting eKP into the approximated polynomial, the knee point strength FKP can be determined. The procedure is graphically shown in Fig. 6. FKP and eKP were 454 MPa and 2.44%, respectively. We performed bearing tests and acquired conventional bearing properties (bearing stiffness Eb, ultimate bearing strength FULT, 2.0% offset strength F2.0%) and the knee point strength FKP simultaneously. Table 1 summarizes the results of the bearing tests (n = 15). All properties seemed to be within a rational scatter for material test results. The scatter of FKP was larger than the other properties but less than 10% C.V. Therefore, the knee point strength FKP can be recognized as an inherent property of composite mechanical joints. The knee point strength FKP was defined as the stress at which nonlinear behavior starts. By acquiring the FKP value, the design process for situations with low-level loads can be made more efficient. 3. Finite element analysis In this section, we describe FE analysis to emulate the damage propagation process in a bearing test specimen. Hashin and LaRC04 criteria were adopted for the analysis. We were able to emulate not only the bearing stress–strain relationship but also the stiffness reduction behavior associated with internal damage propagation.
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The Hashin criteria have four formulae (Eqs. (4)–(7)), whereas the LaRC04 criteria have six (Eqs. (8)–(13)). For a more detailed explanation, see [21,22]. Hashin criteria
Tensile fiber
Compressive fiber Tensile matrix þ
Fig. 5. Bearing stress–strain curves and SCRs of specimens under different load levels.
3.1. Key specifications of the finite element model In this study, FE analysis was performed on ANSYS ver. 13.0. Fig. 7 shows a three-dimensional FE model of the bearing test system. The symmetry about the thickness (z) direction was used for the analysis; therefore, the stacking sequence for the specimen was [45/0/45/90]2. The model consisted of the specimen, a bolt (fastener), and a washer to correspond to the testing jig. A tightening torque by the bolt was introduced as a displacement load given to the washer. The elastic moduli and strengths of the material in this analysis were equal to those of the material used in the experiment section. The values were listed in Table 2 (elastic constants) and Table 3 (strengths and fracture toughnesses) [18,19]. Steel was assumed to be for the fastener and washer material, similar to the experiments (E = 207 GPa and m = 0.30).
1 S2L
rL
2 þ
Lt
1 S2L
ðs2LT þ s2LR Þ ¼ 1
1
rT þ rR > 0
T 2t
ð5Þ 1
ðrT þ rR Þ2 þ
S2T
ðs2TR rT rR Þ
ðs2LT þ s2LR Þ ¼ 1
1 4S2T
ðrT þ rR Þ2 þ
ð6Þ
1 Tc
rT þ rR < 0 1 S2T
"
2
Tc 2ST
ðs2TR rT rR Þ þ
1
# 1 ðrT þ rR Þ
ðs2LT þ s2LR Þ ¼ 1
S2L
ð7Þ
LaRC04 criteria
Tensile matrix LaRC04 #1 ð1 gÞ
!2
rT
rT
Tt
T ist
þg is
þ
sLT
!2 ¼1
SisL
ð8Þ
sT ST gT rn
Compressive matrix LaRC04 #2
2 þ
sL is SL g L r n
!2 ¼1 ð9Þ
Tensile fiber LaRC04 #3
rL LT
¼1
ð10Þ
3.2. Continuum damage mechanics model
Compressive fiber LaRC04 #4 The bearing damage progression in the bearing test specimen was emulated by continuum damage mechanics [2,7,20–23]. This approach requires failure criteria and degradation rules for the elastic properties of the failed elements to judge whether an element in the model fails and to simulate the elastic behavior of damaged materials. Both the Hashin [24] and LaRC04 [25] failure criteria were adopted in the damage propagation analysis for comparison. These criteria classify failure modes and have the same failure modes. However, the number of criterion formulae is different.
ð4Þ
rL < 0 jrL j ¼ Lc
Compressive matrix þ
rL > 0
s1m2m SisL gL r2m2m
¼ 1ðr2m2m < 0Þ ð11Þ
Compressive matrix LaRC04 #5
sTm ST gT rm n
2
sLm is SL gL rm n
þ
!2 ¼1 ð12Þ
Compressive fiber LaRC04 #6 ð1 gÞ
þ
s1m2m SisL
!2
r2m2m T ist
þg
r2m2m
!2
T ist
r2m2m > 0 r11 < L2c
¼1
!
ð13Þ The failure modes and related material property degradation rules are presented in Table 4. The degradation rule intended that if an element failed in fibrous modes, the element loses all of
Table 1 Summary for bearing tests based on ASTM D 5961 Standard.
SD C.V. [%] Fig. 6. How to determine knee point strength FKP.
Stiffness [MPa/%]
Strength [MPa] Ultimate Offset
Knee point
Eb
FULT
F2.0%
FKP
183 6.0 3.0
993 16.6 1.7
919 25.4 1.7
447 37.2 8.3
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Fig. 7. Finite element model of bearing test system.
Table 2 Elastic constants for finite element model of bearing test system. Laminate Moduli
Poisson’s ratio
Longitudinal (L)
Transverse (T, R)
Shearing (LT, LR)
Shearing (TR)
LT, LR
TR
EL [MPa]
ET [MPa]
GL [MPa]
GT [MPa]
mL [–]
mT [–]
139000 Fastener and washer Modulus E [MPa] 207000
8000
4030
2985
0.34
0.34
Poisson’s ratio
m [–] 0.30
Table 3 Strengths and fracture toughnesses for composite laminate in finite element model of bearing test system. Strength
Fracture toughness
Longitudinal
Transverse
Shear
Mode I
Mode II
Tensile
Compressive
Tensile
Compressive
Lt [MPa]
Lc [MPa]
Tt [MPa]
Tc [MPa]
SLT [MPa]
GIc [kJ/m2]
GIIc [kJ/m2]
3100
1242
66.9
146
118
0.54
1.64
Table 4 Degradation Rules for Damage Progression Analysis.
Transverse-tensile Transverse-compressive Fiber-tensile
Fiber-compressive
Degradation rules
Hashin
LaRC04
ET ? 0.2 ET GLT ? 0.2 GLT mLT, mTR, mLR ? 0.0
Eq. (6)
Eq. (8)
Eq. (7)
Eqs. (9) and (12)
EL ? 0.2 EL ET ? 0.2 ET ER ? 0.2 ER mLT,, mTR, mLR ? 0.0 GLT ? 0.2 GLT GTR ? 0.2 GTR GLR ? 0.2 GLR
Eq. (4)
Eq. (10)
Eq. (5)
Eqs. (11) and (13)
load-bearing capacities, else if failed in matrix modes, reinforcing fibers remain in the element, so the element still have a load bearing capacity in longitudinal direction. The reduction factor of 0.2 was carefully selected from some references [2,7,21–23]. The factors for stiffness reduction were used within the range of 0.0–0.3 in the references. The case in which the factor of 0.2 was used showed a good agreement with experimental result visually.
After applying the reduction factor, redistribution of the internal load was considered. The displacement and stress fields were recalculated till the damage progression converges. In addition, the ‘‘in situ strengths’’ in [25] were adopted for the plies in a laminate instead of the UD composite test results for both the Hashin and LaRC04 criteria. Bearing stress–strain (rb–eb) curves obtained by FE analysis and experimentally are compared in Fig. 8. The figure also shows curves of the SCRs. The increment of enforced displacement was a constant 0.02 mm. Both the rb–eb and SCR curves showed good agreement. The damage states of 0° layers just before and after knee points, which are the first local minima on the SCR curves, are shown in Fig. 9 (Hashin criteria) and Fig. 10 (LaRC04). Fig. 9 suggests that the stiffness reduction after the knee point was involved with fiber compressive failures. Fig. 10 appears to show two local minima on the analytical SCR curve. Therefore, the damage states were present with respect to the first and second knee points. According to the LaRC04 criteria, the fiber compressive failures propagated gradually along the fastener hole edge between the first and
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M. Nakayama et al. / Composite Structures 132 (2015) 1–11
be the bearing stress when an apparent reduction in bearing stiffness (3.0% reduction compared to the former load step) occurs. Fig. 11 shows rb–eb curves by the Hashin and LaRC04 criteria for the estimated knee point strengths. To detect a knee point on the stress–strain curve precisely, the increment of displacement was varied according to predetermined values (0.020, 0.0028, 0.0004 mm). Based on the Hashin criteria, the estimated knee point strength was 520 MPa; based on the LaRC04 criteria, the estimated strength was 360 MPa. The mean FKP in the experiment was 447 MPa (see Table 1); therefore, the Hashin criteria seem to overestimate FKP, and the LaRC04 criteria underestimate it. 4. Determination of design factor One of the problems with composite structures is the excessively conservative design. Large design factors are multiplied with the external load conditions. To avoid conservative designs, the design factors need to be optimized. In this section, we propose reliability-based determination for the design factor based on the FE analysis presented in Section 3. We previously proposed a method to acquire reliability-based design factors in [17]. The method was also adopted in the present study.
Fig. 8. Comparison of experimental and analytical bearing behaviors.
second knee points. A severe reduction in bearing stiffness occurred at the second knee, and the damage extended to inside the specimen. According to the FE analysis results, failures in 0° layers caused the stiffness reductions; this coincides with the prediction based on microscope observation.
4.1. Design parameters
3.3. Prediction for knee point strength
The damage propagation analysis in the previous section had many design variables such as material properties (e.g., strength and elastic constant) and dimension parameters (e.g., clearance between fastener and hole and ply misalignment). Considering all of these variables requires a prohibitive amount of computing cost. To reduce the cost, design variables which significantly affect the knee point strength FKP should be chosen. Some feasibility analyses were performed using the experimental design method and
In Section 2, FKP was regarded as the starting point of apparent nonlinearity on the bearing stress–strain curve. The SCR was developed to detect the knee point. In contrast to noisy experimental data, the FE analysis results seemed little affected by noise. Therefore, we were able to detect the knee point on the analytical rb–eb curves directly. We predicted the knee point strength FKP to
INNER
OUTER
INNER
Before the knee Point
OUTER After the knee point
1400 1200 1000
Transverse-compressive Fiber-tensile Fiber-compressive
800 600 400 200 0 -1
0
1
2
3
4
5
Fig. 9. Damage progression in 0° plies through the knee point (Hashin criterion).
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INNER
OUTER
INNER
OUTER
Before the first knee point
INNER
OUTER
After the first knee point
INNER
Before the second knee point
OUTER After the second knee point
1400 1200
#2: Transverse-compressive #4: Fiber-compressive
1000 800 600
#5: Transverse-compressive
400
#6: Fiber-compressive
200 0 -1
0
1
2
3
4
5
6
Fig. 10. Damage progression in 0° plies through the knee point (LaRC04 criterion).
Fig. 11. Knee point strength FKP estimation by Hashin and LaRC04 criteria.
analysis of variance. Ten variables, modulus in fiber direction EL, that in transverse direction, ET, Poisson’s ratio m (assumed m = mLT = mTR = mLR for computing cost saving), longitudinal shear
modulus GL, compressive strength in longitudinal direction Lc, critical energy release rate in mode I GIc, compressive strength in transverse direction Tc, critical energy release rate in mode II GIIc,
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M. Nakayama et al. / Composite Structures 132 (2015) 1–11
coefficient for non-linear term in in-plane shear behavior b (contained in SisL implicitly [25]) and fracture angle by transverse compressive load a0 (contained in ST implicitly [25]) were considered. Three variables were screened from the other ones: Lc, Tc, and ET. Other variables were ignored because of negligible effects on the bearing failure behavior and/or their small variability. In both criterion cases, null hypotheses about the three variables were rejected, but the F-statistics of Lc were distinctive; this means that Lc more strongly affected the predictions for FKP than the other parameters. These were called ‘‘design parameters.’’
Table 6 Coefficients and t-statistics for the response surface equation (Eq. (16)) (Unit: [MPa], Hashin Criterion Used).
Coefficients t-Statistics
X ¼ 2ðx lx Þ=d
ð14Þ
where lx is the mean value of parameter x and d is the difference between the maximum and minimum x, respectively. All variables were intended to have a closed interval [1.0, +1.0] through this conversion. Based on the computed results, the form of RS should be nonlinear. Thus, we assumed its form to be quadratic:
F KP ¼ a0 þ a1 Lc þ a2 ET þ a3 T c þ a4 Lc ET þ a5 Lc T c þ a6 ET T c 2 2 2 2 þ a7 L2 c þ a8 ET þ a9 T c þ a10 Lc ET þ a11 Lc T c 2 2 2 þ a12 ET L2 c þ a13 ET T c þ a14 T c Lc þ a15 T c ET
ð15Þ
When the Hashin criteria were used, Eq. (15) was optimized into Eq. (16), and the equation was named F HKP . All the coefficients were determined to maximize the adjusted coefficient of determination R2ad .
R2ad
Their values and t-statistics are listed in Table 6. was more than 99% and seemed to approximate the FE failure analysis well.
F HKP ¼ a0 þ a1 Lc þ a2 ET þ a3 T c þ a4 Lc ET þ a5 Lc T c þ a7 L2 c þ a10 Lc E2 T
ð16Þ
Based on the LaRC04 criteria, Eq. (15) was optimized into Eq. (17) (F LKP ). All the coefficients were determined in the same manner as with the Hashin criteria. Their values and t-statistics are also listed Table 5 Levels for design parameters.
Lv. 1 Lv. 2 Lv. 3
a1
a2
a3
a4
a5
a7
a10
498.2 302.7
123.8 71.9
11.53 6.27
5.05 4.03
1.67 1.79
4.12 4.41
4.54 3.44
2.47 1.52
Table 7 Coefficients and t-statistics for the response surface equation (Eq. (17)) (Unit: [MPa], LaRC04 criterion used).
4.2. Monte Carlo simulation 4.2.1. Response surface method The design parameters were given values called ‘‘levels.’’ First, they were assumed to obey a normal distribution with a mean value equivalent to the experimental value and 5% coefficients of variance (C.V.). The minimum and maximum levels of the parameters were intended to correspond to l 3r and l + 3r, respectively (l = mean value and r = standard deviation). The levels for the three parameters Lc, Tc, and ET are listed in Table 5. They have three levels each, so we were able to generate 33 = 27 combinations; these were substituted into the FE damage analysis presented previously, and we obtained 27 results for each failure criterion. According to the Hashin criteria, the minimum and maximum results were 347.4 and 639.3 MPa, respectively. According to the LaRC04 criteria, these results were 276.1 and 461.9 MPa, respectively. Based on the previous 27 2 results for the failure analysis, we can estimate the response surface (RS) equations that emulate the behavior of the analysis on the variance of the design parameters. For computing accuracy, the normalized design parameter x⁄ was introduced as follows:
a0
Coefficients t-Statistics
a0
a1
a2
a3
a7
a12
363.5 93.5
66.18 17.1
16.50 3.59
1.64 1.96
8.04 2.57
11.63 3.04
in Table 7. The adjusted coefficient of determination R2ad was more than 99%. 2 F LKP ¼ a0 þ a1 Lc þ a2 ET þ a3 T c þ a7 L2 c þ a12 ET Lc
ð17Þ
A comparison of F HKP (Eq. (16)) and F LKP (Eq. (17)) shows that F LKP has fewer terms than F HKP . In particular, coupled terms of Lc* and other variables seem to have been omitted. This is because of the difference in failure criteria. The Hashin criteria contain Lc in explicit form (Eq. (5)), but the LaRC04 criteria do not. Lc is only used as a threshold for LaRC04 #6 (Eq. (13)). Therefore, the LaRC04 criteria are less affected by Lc than the Hashin criteria. Fiber-compressive failure is a significant failure mode for the FKP prediction. As discussed in a later section, this produces a difference between reliability-based design factors. 4.2.2. Monte Carlo simulation Monte Carlo simulation (MCS) was performed to reveal the stochastic properties of the knee point strength of bolted joints in CFRP laminates. The MCS took 20,000 iterations to fully converge. The design parameters were regarded as probabilistic variables assumed in the former section. Thus, the normalized parameters, Lc ; ET , and T c were considered to be probabilistic variables obeying normal distributions with a mean value of 0.0 and standard deviation of 0.33. Fig. 12 shows the MCS results in the form of a histogram and probability density function. The coefficient of variance for the design parameters was 10%. The applied distributions were verified by v2 test. When the Hashin criteria were used (Fig. 12(a)), the results could be applied into a normal distribution (Mean: 497 MPa, SD: 41.3 MPa). On the other hand, the results from the LaRC04 criteria (Fig. 12(b)) were applied into a log-normal distribution (Mean: 364 MPa, SD: 22.7 MPa). The difference was because of the forms of the response surface equations (see Eqs. (16) and (17)). Eq. (16) (Hashin, F HKP ) seemed to have stronger linearity with respect to the design parameters than Eq. (17) (LaRC04, F LKP ). The linear sums of the normal probabilistic variables were also normal probabilistic variables. As previously noted, the Hashin criteria seemed to overestimate FKP, and the LaRC04 criteria underestimated it. According to the experimental results for FKP, its mean value was 447 MPa and the standard deviation was 37.2 MPa (8.3% C.V.). Numerical results suggested standard deviations of 41.3 MPa (8.3% C.V.) and 22.7 MPa (6.2% C.V.) according to the Hashin and LaRC04 criteria; these seem reasonable compared to the experimental values. 4.3. Design factor by probabilistic method
Lc [MPa]
ET [MPa]
Tc [MPa]
1056 1242 1428
6800 8000 9200
124 146 168
4.3.1. Limit stress based on reliability Fig. 13 compares limit stresses—defined as the stress generated when a limit load is applied to the structure[14]—according to the
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M. Nakayama et al. / Composite Structures 132 (2015) 1–11
Fig. 12. Results of Monte Carlo simulations. (a) Distribution for FKP (Hashin criteria used: 10% C.V.). (b) Distribution for FKP (LaRC04 criteria used: 10% C.V.).
probabilistic and deterministic designs. Fig. 13(a) shows the case with the Hashin criteria, and Fig. 13(b) shows the case with the LaRC04 criteria. The graphs show the probability of failure (POF) and applied bearing stress rb as the vertical and horizontal axes, respectively. The curve is the cumulative density function of FKP found by integration of the probability density functions, and the limit stress by probabilistic design was determined as follows. Three assumptions were made: (1) the applied stress was constant (not a probabilistic variable); (2) the allowable probability of failure was 108; and (3) the margin of safety (MS) was equal to zero. The MS indicates the margin between a load applied to the structure and the allowable load. This value was assumed to be zero:
MS ¼
PA 1 ¼ 0 ) PA ¼ DF PLimit DF PLimit
ð18Þ
where PA is the allowable load for the structure, DF is the design factor, and PLimit is the limit load for the structure. The limit stress is thus introduced as follows [14]:
rA ¼ DF rLimit
ð19Þ
where rA is the allowable stress and rLimit is the limit stress. Based on these assumptions, the stress when the probability of failure is equal to 108 on the cumulative density function was estimated to be 265 MPa (Hashin) and 256 MPa (LaRC04). These values correspond to the probabilistic limit stresses. In deterministic design, the limit stress is determined based on the allowable stress by FE analysis and the fitting factor. FE analysis estimated FKP to be 399 MPa (Hashin) and 312 MPa (LaRC04) when the design
Fig. 13. Comparison between probabilistic and deterministic designs of limit stress for bolted joint in composite laminate (10% C.V.) (a) Hashin criteria: 10% C.V. (b) LaRC04 criteria: 10% C.V.
parameters used A-basis [11] values as their design tolerances. The design tolerance was statistically acquired from the following definition. ‘‘A-basis’’ indicates that at least 99% of the population was expected to equal or exceed the statistically calculated mechanical property value with a confidence of 95%. However, FKP should be regarded as containing a margin derived from the fitting factor. The fitting factor is multiplied with the applied load condition, and the load multiplied by the fitting factor must be lower than the strength of the joint. Therefore, FKP from FE analysis was divided by the fitting factor, which was assumed to be 1.15. Finally, the limit stress for the mechanical joint by the deterministic method was 347 (Hashin) and 272 MPa (LaRC04). The values from the probabilistic method were less than the deterministic values. Not only the failure criteria but also the coefficients of variances affected the results. Fig. 14 compares the probabilistic and deterministic limit stresses when 5% coefficients of variance for the design parameters were assumed. The computed limit stresses seemed to almost correspond to each other. 4.3.2. Reliability-based design factor The results from the probabilistic design method can be converted into the form of a design factor. FKP from FE analysis can be considered to be equal to the product of the probabilistic design factor and the probabilistic limit stress. Therefore, the probabilistic limit stresses under previous conditions can be converted into the fitting factors; these values are presented in Table 8. The conventional design for a mechanical joint requires a fitting factor of more than 1.15. The reliability-based factors are close to the conventional values. However, the value of 1.15 is set for metallic
M. Nakayama et al. / Composite Structures 132 (2015) 1–11
11
stress–strain curve; our research related the nonlinearity, especially in the early steps in the bearing curve, to internal damages (compressive damage in the 0° layers) in the bearing-loaded specimens. This suggestion was supported by microscope observation and FE analysis. The response surface method was used to perform MCS and acquire the stochastic properties of the knee point strength. Designs (limit stresses) by probabilistic and deterministic methods were compared, and the probabilistic design was converted into the form of a fitting factor, and its proper value was proposed. References
Fig. 14. Comparison between probabilistic and deterministic designs of limit stress for bolted joint in composite laminate (5% C.V.) (a) Hashin criteria: 5% C.V. (b) LaRC04 criteria: 5% C.V.
Table 8 Design factors based on failure probability of 108. C.V. (%)
Hashin
LaRC04
10 5
1.51 1.18
1.22 1.12
structures; therefore, actual composite structures have higher values. Thus, the reliability-based design factors in this study seem to be less than the actual values. The design factors vary according to the assumed variances of material properties. The results suggest that composite structures can be refined and made more efficient by evaluation of the variances of the design parameters and factoring them into the design factors. 5. Conclusion A design-oriented strength for bolted joints in CFRP laminates was proposed and analyzed probabilistically. The strength FKP was defined as the stress value at the knee point on the bearing
[1] Irisarri F-X, Laurin F, Carrere N, Maire J-F. Progressive damage and failure of mechanically fastened joints in CFRP laminates – Part I: Refined finite element modelling of single-fastener joints. Compos Struct Jul. 2012;94(8):2269–77. [2] Hühne C, Zerbst a-K, Kuhlmann G, Steenbock C, Rolfes R. Progressive damage analysis of composite bolted joints with liquid shim layers using constant and continuous degradation models. Compos Struct 2010;92(2):189–200. Jan. [3] Atasß A, Mohamed GF, Soutis C. Effect of clamping force on the delamination onset and growth in bolted composite laminates. Compos Struct 2012;94(2):548–52. [4] Aktasß A. Bearing strength of carbon epoxy laminates under static and dynamic loading. Compos Struct Mar. 2005;67(4):485–9. [5] Crews JH, Naik RA. Bearing-bypass loading on bolted composite joints. Nasa technical memorandum 89153; 1987. [6] Xiao Y, Ishikawa T. Bearing strength and failure behavior of bolted composite joints (Part I: Experimental investigation). Compos Sci Technol Jun. 2005;65(7–8):1022–31. [7] Xiao Y, Ishikawa T. Bearing strength and failure behavior of bolted composite joints (Part II: Modeling and simulation). Compos Sci Technol Jun. 2005;65(7– 8):1032–43. [8] Atas A, Soutis C. Subcritical damage mechanisms of bolted joints in CFRP composite laminates. Compos Part B: Eng 2013;54:20–7. [9] Atas A, Soutis C. Strength prediction of bolted joints in CFRP composite laminates using cohesive zone elements. Compos Part B: Eng 2014;58:25–34. [10] ASTM D 5961/D 5961M Standard test method for bearing response of polymer matrix composite laminates; 2007. [11] Certification of transport airplane structure, FAA Advisory Circular AC25-21; 1999. [12] Crews JH, Naik RA. Failure analysis of a graphite/epoxy laminate subjected to bolt bearing loads. NASA TM 86297; 1984. [13] Crews JH, Naik RA. Combined bearing and bypass loading on a graphite/epoxy laminate. Compos Struct 1986;6:21–40. [14] Long MW, Narciso JD. Probabilistic design methodology for composite aircraft structures. DOT/FAA/AR-99/2; 1999. [15] Chamis CC. Probabilistic simulation of multi-scale composite behavior. Theor Appl Fract Mech 2004;41(1–3):51–61. [16] Li H, Lu Z, Zhang Y. Probabilistic strength analysis of bolted joints in laminated composites using point estimate method. Compos Struct 2009;88(2):202–11. [17] Nakayama M, Uda N, Ono K. Probabilistic assessment of pin joint strength in CFRP laminates. Compos Struct 2012;93(8):2026–30. [18] JAXA Advanced Composites Data Base, JAXA-ACDB. [Online]. Available from:. [19] Yokozeki T, Ogasawara T, Ishikawa T. Nonlinear behavior and compressive strength of unidirectional and multidirectional carbon fiber composite laminates. Compos Part A: Appl Sci Manuf 2006;37(11):2069–79. [20] Matzenmiller A, Lubliner J, Taylor RL. A constitutive model for anisotropic damage in fiber-composites. Mech Mater 1995;20(2):125–52. [21] Dano M-L, Kamal E, Gendron G. Analysis of bolted joints in composite laminates: strains and bearing stiffness prediction. Compos Struct 2007;79(4):562–70. [22] Chang FK, Chang KY. A progressive damage model for laminated composites containing stress concentrations. J Compos Mater 1987;21(9):834–55. [23] Tan SC. A progressive failure model for composites laminate containing openings. J Compos Mater 1991;25(5):556–77. [24] Hashin Z. Failure criteria for unidirectional fiber composites. J Appl Mech 1980;47:329–34. [25] Pinho ST, Dávila CG, Camanho PP, Iannucci L, Robinson P. Failure models and criteria for FRP under in-plane or three-dimensional stress states including shear nonlinearity. NASA/TM-2005-213530; 2005.
Contents lists available at ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Design-oriented strength of mechanical joints in composite laminate structures and reliability-based design factor Masahiro Nakayama a,⇑, Nobuhide Uda a,1, Kousei Ono a,1, Shin-ichi Takeda b, Tetsuya Morimoto b a b
Department of Aeronautics and Astronautics, Kyushu University, 744 Moto-oka, Nishi-ku, Fukuoka 819-0395, Japan Japan Aerospace Exploration Agency, 6-13-1 Osawa, Mitaka-shi, Tokyo 181-0015, Japan
a r t i c l e
i n f o
Article history: Available online 2 May 2015 Keywords: Composite material Mechanical joint Design Strength Bearing test
a b s t r a c t Mechanical joints may act as bottlenecks in the design process of composite structures. Fitting factors need to be multiplied with the applied load condition to assure reliability. For more efficient design, this paper proposes a brand-new design-oriented strength: knee point strength for the mechanical joints in composite laminates. The knee point strength is derived from the stiffness change rate for the bearing stress–strain behavior in a bearing test. The strength can be correlated with the internal bearing damage; this is verified by micrography about the bearing test specimen and finite element analysis. The probabilistic design and analysis method was applied to acquire the reliability-based fitting factor. An approach combining finite element damage analysis and a stochastic technique was adopted to analyze the probabilistic properties for the design-oriented strength of the bolted joints in CFRP laminates. The probabilistic and deterministic designs were compared and discussed. Then a proper value of the fitting factor was proposed. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Applications of advanced composite materials such as carbon-fiber-reinforced plastics (CFRPs) are expanding in not only the aerospace industry but also other industries because of their high specific strength and stiffness, good fatigue resistance, etc. However, composite materials generally have more scattering properties than conventional metallic materials. Therefore, the ‘‘design tolerance,’’ which is the material strength used in structural design processes, is reduced. The tolerances for composite materials need to be set low because of the large variation in their strength properties. In addition, there is much less experience with using composites as structural materials than with metals. Therefore, structural designs that use composite materials tend to be conservative. One example of conservative structural elements is mechanical joints. Composite-made joints have often been the subjects of many studies both experimentally and analytically [1–9]. In actual applications, mechanical joints are commonly used to assemble structural components; they are necessary because they allow re-access to internal structures and easy replacement of damaged
⇑ Corresponding author. Tel./fax: +81 92 802 3036. 1
E-mail address: [email protected] (M. Nakayama). Tel./fax: +81 92 802 3036.
http://dx.doi.org/10.1016/j.compstruct.2015.04.044 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.
components. They can be critical to strength assessment of the structure because of the stress concentrations their opening holes induce. ASTM standard D 5961 [10] defines the bearing strengths under high loads, which corresponds to the ultimate load state. The testing standard does not define the strengths under relatively low applied loads, which correspond to the limit load state. The ‘‘fitting factor’’ is a design factor with regard to airworthiness [11]; it is multiplied with external loads separately from the safety factor. The fitting factor is greater than or equal to 1.15 [11]. However, the value for joints in CFRP laminate structures should be higher than 1.15 because of the previously noted uncertainty and lack of empirical knowledge. Therefore, more reasonable definitions for the design strength and a suitable design factor are desirable for efficient design of mechanical joints in CFRP structures. This paper proposes a brand-new design-oriented strength for mechanical joints in composite laminates and a reasonable design factor for the joints. Two approaches are adopted. Knee point strength FKP is a design-oriented strength based on an experimental approach. The strength is acquired by focusing on stiffness reduction during bearing behavior, which is unfavorable for actual structures. Former researches about the bearing strength [2,12,13] seem that they focus on nonlinearity of bearing load-displacement relationship. Our research related the nonlinearity, especially in the early steps in the bearing curve, to internal damages in the
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bearing-loaded specimens. We developed a method to identify the starting point of the stiffness reduction, which would be used as a criterion for the limit stress, defined as the stress generated in the composite joint when a limit load is applied to the structure [14]. This study utilized micrography and damage propagation analysis by finite element method to reveal the relationship between the internal bearing damage and the stiffness reduction. The second analytical approach sets the fitting factor based on the reliability of the joint. To obtain greater efficiency in the design of structures made from composites, attempts have been made to introduce a probabilistic or reliability-based design method [15– 17]; this is generally called the probabilistic design and analysis (PDA) method. This method optimizes structures on the basis of their probability of failure or reliability; it should be able to avoid too conservative designs and more fully utilize the good mechanical properties of composite materials. Damage propagation analysis using the finite element model to calculate FKP was combined with the Monte Carlo method to estimate the stochastic properties
of the design strength. Allowable stresses by probabilistic and deterministic designs were compared, and the probabilistic one was translated into the form of fitting factor. The PDA method can be used for providing reasonable fitting factors for composite structures. 2. Experimental 2.1. Bearing test The bearing test is a standard method for evaluating the properties of composite joints. Fig. 1 shows a simplified image of the test and a photograph of an actual system. The CFRP specimen had a fastener hole, and a bearing load was applied through the fastener. Extensometers were equipped on both edges of the specimen to measure relative displacements between the specimen and testing jig. In this study, Toray T800S/3900-2B [18,19] prepreg was used to fabricate the laminates. The composite laminate
Fig. 1. Schematic and actual photograph for the bearing test. (a) Schematic for bearing test. (b) Actual photograph.
M. Nakayama et al. / Composite Structures 132 (2015) 1–11
3
stacking sequence was [45/0/45/90]2S (quasi-isotropic); therefore, the thickness of the laminate was about 3.0 mm. Failure modes of the bearing test specimens [10] vary depending on stacking sequence of the specimens, dimensions of the specimens, and so on. The stacking sequence of our specimens was decided to cause failures in bearing mode (not shear-out or net tension modes). Fig. 2 shows the dimensions of the bearing test specimen. The solid curve in Fig. 3 is a typical bearing stress–strain relationship for the bearing behavior. The zigzag curve in the upper zone of the figure represents the stiffness change rate (SCR) for the bearing behavior; the SCR is explained in the next section. The bearing stress rb and strain eb are defined as follows [10]:
rb ¼ P=ðD hÞ eb ¼
d1 þ d2 2D
ð1Þ ð2Þ Fig. 3. Typical bearing stress–strain diagram.
where P is a load that is applied to the test system, D is the diameter of the fastener hole in the laminate, h is the thickness of the laminate, and di (i = 1, 2) are displacements measured by each of the two extensometers. The stress rb rapidly rose at first because of static friction between the testing jigs and specimen fastened to them. The bearing load increased to a certain degree; the specimen then slipped in the jigs, and the specimen and fastener bolt came into contact. The contact area then bore most of the load. The bearing stress increased linearly initially; after a certain stress level (i.e., defined as the ‘‘knee point’’ in this paper), the curve showed nonlinear behavior. During the nonlinear behavior, internal damage proceeded; the curve finally reached the maximum bearing stress, which is called the ultimate bearing strength Fb in the testing standard [10]. The term ‘‘knee point’’ suggests the starting point of the nonlinear bearing behavior. 2.2. Stiffness change rate This study focused on the change in stiffness of the bearing behavior. The stiffness decreased gradually during the bearing load process. This reduction can be correlated with the internal damage. The zigzag curve in Fig. 3 shows a second-order differential of the rb–eb curve, which corresponds to the rate of change in the bearing stiffness and is hereafter called the SCR:
SCR ¼
@ 2 rb @ e2b
ð3Þ
The SCR was plotted based on sampled data of the bearing test. The data were acquired in discrete form; therefore, the differential should be converted into the central difference. The absolute values of the SCR seem to be irrelevant. Thus, the vertical axis was not
included in the diagram. However, the scales for all SCRs (include later ones) should be standardized. The differences were strongly affected by the sampling rate of the data logger and signal noise. The sampling rate was 2.0 Hz, and the data points were thinned to have an interval of 0.2% bearing strain to remove the effect of the noise. These values were determined empirically. Then second central difference calculations were performed based on the thinned data points and the result was plotted in Fig. 3 as the upper zigzag curve. The curve had some local minima, which suggests that the bearing stress–strain curve had some ‘‘knees.’’ These were assumed to be induced by damage to the laminate. Fig. 4 shows the micrographs of the cross section of the specimens with different states of damage. These specimens are different from the one used to draw the curve in Fig. 3, but all tests were conducted under similar conditions. The specimens shown in Fig. 4 were loaded until their bearing strength reached 270, 380, 490, and 590 MPa. Fig. 5 shows the rb–eb curves and the SCRs of the specimens. However, the 270 and 380 MPa specimens do not have SCRs because of their insufficient data points. The SCRs of the other more highly loaded specimens suggest that the 270 and 380 MPa specimens had positive or slightly negative SCR values. For the 270 MPa bearing stress, where a local minimum did not appear on the SCR curve, no damage was observed in 0° plies, as shown in Fig. 4 (the chip on the left-hand side was made when the specimen was sanded for microscope observation). At 380 MPa, the outer 0° layers had compressive failures near the bearing-loaded areas. The failures were observed to propagate with the bearing load (see photographs for 490 and 590 MPa). Therefore, stiffness reduction is involved with the compressive damages in 0° layers; this result was supported by FE analysis and is later discussed in Section 3.
D = 6.00 + 0.03 – 0.00 mm W = 36 mm L = 135 mm h = 3.0 mm e = 18.0 mm
Fig. 2. Dimensions of bearing test specimen.
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-45
0 45 90 -45 0
45 90 90 45 0 -45 90 45 0 -45
270 MPa
-45 0 45 90 -45
0 45 90 90 45 0 -45 90 45 0 -45
380 MPa
-45 0 45 90 -45
0 45 90 90 45 0 -45 90 45 0 -45
-45 0 45 90 -45
0 45 90 90 45 0 -45 90 45 0 -45
490 MPa
590 MPa
Fig. 4. Micrographs of cross sections on the bearing load line in specimens.
2.3. Knee point on the bearing stress–strain curve The testing standard ASTM D5961 [10] defines two types of bearing strength: (1) the maximum value of the bearing stress Fb and (2) the stress at an intersection of the rb–eb curve, and a line whose slope is equal to the bearing stiffness Eb given a 2.0% offset strain. The strength is called the 2.0% offset strength F2.0. In addition, some researchers have proposed more physical-based definitions of the bearing strength [2,4,13]. However it seemed that they used the nonlinearity as just the threshold for the bearing strength determination. The nonlinearity was not related to the internal bearing damages. For an actual structure, a reduction in stiffness before an applied load reaches its limit value is undesirable. Thus, the knee point strength (hereafter referred to as FKP), which is the stress of the first local minimum of the second-order differential of the rb–eb curve, is proposed as the design strength for bolted joints in composite laminates. FKP is simple to detect, and there is no need for further experimental equipment. It provides a more physical-based design strength value than conventional alternatives. However, there are issues regarding the accuracy in determining the knee point on the rb–eb curve. We developed a method that uses the fourth-order polynomial approximation of the rb–eb curve. First, based on the SCR curve calculated from thinned data points, a temporary knee point is determined. Next, a region near the temporary knee point on the rb–eb curve is approximated by a fourth-order polynomial. The range of ±30 MPa around the temporary knee point was empirically found to be suitable for the
approximated region. The second-order differential of the polynomial suggests the SCR of the approximated region, and the curve of the second-order differential must have a local minimum. The strain value at the local minimum is the knee point strain eKP. By substituting eKP into the approximated polynomial, the knee point strength FKP can be determined. The procedure is graphically shown in Fig. 6. FKP and eKP were 454 MPa and 2.44%, respectively. We performed bearing tests and acquired conventional bearing properties (bearing stiffness Eb, ultimate bearing strength FULT, 2.0% offset strength F2.0%) and the knee point strength FKP simultaneously. Table 1 summarizes the results of the bearing tests (n = 15). All properties seemed to be within a rational scatter for material test results. The scatter of FKP was larger than the other properties but less than 10% C.V. Therefore, the knee point strength FKP can be recognized as an inherent property of composite mechanical joints. The knee point strength FKP was defined as the stress at which nonlinear behavior starts. By acquiring the FKP value, the design process for situations with low-level loads can be made more efficient. 3. Finite element analysis In this section, we describe FE analysis to emulate the damage propagation process in a bearing test specimen. Hashin and LaRC04 criteria were adopted for the analysis. We were able to emulate not only the bearing stress–strain relationship but also the stiffness reduction behavior associated with internal damage propagation.
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The Hashin criteria have four formulae (Eqs. (4)–(7)), whereas the LaRC04 criteria have six (Eqs. (8)–(13)). For a more detailed explanation, see [21,22]. Hashin criteria
Tensile fiber
Compressive fiber Tensile matrix þ
Fig. 5. Bearing stress–strain curves and SCRs of specimens under different load levels.
3.1. Key specifications of the finite element model In this study, FE analysis was performed on ANSYS ver. 13.0. Fig. 7 shows a three-dimensional FE model of the bearing test system. The symmetry about the thickness (z) direction was used for the analysis; therefore, the stacking sequence for the specimen was [45/0/45/90]2. The model consisted of the specimen, a bolt (fastener), and a washer to correspond to the testing jig. A tightening torque by the bolt was introduced as a displacement load given to the washer. The elastic moduli and strengths of the material in this analysis were equal to those of the material used in the experiment section. The values were listed in Table 2 (elastic constants) and Table 3 (strengths and fracture toughnesses) [18,19]. Steel was assumed to be for the fastener and washer material, similar to the experiments (E = 207 GPa and m = 0.30).
1 S2L
rL
2 þ
Lt
1 S2L
ðs2LT þ s2LR Þ ¼ 1
1
rT þ rR > 0
T 2t
ð5Þ 1
ðrT þ rR Þ2 þ
S2T
ðs2TR rT rR Þ
ðs2LT þ s2LR Þ ¼ 1
1 4S2T
ðrT þ rR Þ2 þ
ð6Þ
1 Tc
rT þ rR < 0 1 S2T
"
2
Tc 2ST
ðs2TR rT rR Þ þ
1
# 1 ðrT þ rR Þ
ðs2LT þ s2LR Þ ¼ 1
S2L
ð7Þ
LaRC04 criteria
Tensile matrix LaRC04 #1 ð1 gÞ
!2
rT
rT
Tt
T ist
þg is
þ
sLT
!2 ¼1
SisL
ð8Þ
sT ST gT rn
Compressive matrix LaRC04 #2
2 þ
sL is SL g L r n
!2 ¼1 ð9Þ
Tensile fiber LaRC04 #3
rL LT
¼1
ð10Þ
3.2. Continuum damage mechanics model
Compressive fiber LaRC04 #4 The bearing damage progression in the bearing test specimen was emulated by continuum damage mechanics [2,7,20–23]. This approach requires failure criteria and degradation rules for the elastic properties of the failed elements to judge whether an element in the model fails and to simulate the elastic behavior of damaged materials. Both the Hashin [24] and LaRC04 [25] failure criteria were adopted in the damage propagation analysis for comparison. These criteria classify failure modes and have the same failure modes. However, the number of criterion formulae is different.
ð4Þ
rL < 0 jrL j ¼ Lc
Compressive matrix þ
rL > 0
s1m2m SisL gL r2m2m
¼ 1ðr2m2m < 0Þ ð11Þ
Compressive matrix LaRC04 #5
sTm ST gT rm n
2
sLm is SL gL rm n
þ
!2 ¼1 ð12Þ
Compressive fiber LaRC04 #6 ð1 gÞ
þ
s1m2m SisL
!2
r2m2m T ist
þg
r2m2m
!2
T ist
r2m2m > 0 r11 < L2c
¼1
!
ð13Þ The failure modes and related material property degradation rules are presented in Table 4. The degradation rule intended that if an element failed in fibrous modes, the element loses all of
Table 1 Summary for bearing tests based on ASTM D 5961 Standard.
SD C.V. [%] Fig. 6. How to determine knee point strength FKP.
Stiffness [MPa/%]
Strength [MPa] Ultimate Offset
Knee point
Eb
FULT
F2.0%
FKP
183 6.0 3.0
993 16.6 1.7
919 25.4 1.7
447 37.2 8.3
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M. Nakayama et al. / Composite Structures 132 (2015) 1–11
Fig. 7. Finite element model of bearing test system.
Table 2 Elastic constants for finite element model of bearing test system. Laminate Moduli
Poisson’s ratio
Longitudinal (L)
Transverse (T, R)
Shearing (LT, LR)
Shearing (TR)
LT, LR
TR
EL [MPa]
ET [MPa]
GL [MPa]
GT [MPa]
mL [–]
mT [–]
139000 Fastener and washer Modulus E [MPa] 207000
8000
4030
2985
0.34
0.34
Poisson’s ratio
m [–] 0.30
Table 3 Strengths and fracture toughnesses for composite laminate in finite element model of bearing test system. Strength
Fracture toughness
Longitudinal
Transverse
Shear
Mode I
Mode II
Tensile
Compressive
Tensile
Compressive
Lt [MPa]
Lc [MPa]
Tt [MPa]
Tc [MPa]
SLT [MPa]
GIc [kJ/m2]
GIIc [kJ/m2]
3100
1242
66.9
146
118
0.54
1.64
Table 4 Degradation Rules for Damage Progression Analysis.
Transverse-tensile Transverse-compressive Fiber-tensile
Fiber-compressive
Degradation rules
Hashin
LaRC04
ET ? 0.2 ET GLT ? 0.2 GLT mLT, mTR, mLR ? 0.0
Eq. (6)
Eq. (8)
Eq. (7)
Eqs. (9) and (12)
EL ? 0.2 EL ET ? 0.2 ET ER ? 0.2 ER mLT,, mTR, mLR ? 0.0 GLT ? 0.2 GLT GTR ? 0.2 GTR GLR ? 0.2 GLR
Eq. (4)
Eq. (10)
Eq. (5)
Eqs. (11) and (13)
load-bearing capacities, else if failed in matrix modes, reinforcing fibers remain in the element, so the element still have a load bearing capacity in longitudinal direction. The reduction factor of 0.2 was carefully selected from some references [2,7,21–23]. The factors for stiffness reduction were used within the range of 0.0–0.3 in the references. The case in which the factor of 0.2 was used showed a good agreement with experimental result visually.
After applying the reduction factor, redistribution of the internal load was considered. The displacement and stress fields were recalculated till the damage progression converges. In addition, the ‘‘in situ strengths’’ in [25] were adopted for the plies in a laminate instead of the UD composite test results for both the Hashin and LaRC04 criteria. Bearing stress–strain (rb–eb) curves obtained by FE analysis and experimentally are compared in Fig. 8. The figure also shows curves of the SCRs. The increment of enforced displacement was a constant 0.02 mm. Both the rb–eb and SCR curves showed good agreement. The damage states of 0° layers just before and after knee points, which are the first local minima on the SCR curves, are shown in Fig. 9 (Hashin criteria) and Fig. 10 (LaRC04). Fig. 9 suggests that the stiffness reduction after the knee point was involved with fiber compressive failures. Fig. 10 appears to show two local minima on the analytical SCR curve. Therefore, the damage states were present with respect to the first and second knee points. According to the LaRC04 criteria, the fiber compressive failures propagated gradually along the fastener hole edge between the first and
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be the bearing stress when an apparent reduction in bearing stiffness (3.0% reduction compared to the former load step) occurs. Fig. 11 shows rb–eb curves by the Hashin and LaRC04 criteria for the estimated knee point strengths. To detect a knee point on the stress–strain curve precisely, the increment of displacement was varied according to predetermined values (0.020, 0.0028, 0.0004 mm). Based on the Hashin criteria, the estimated knee point strength was 520 MPa; based on the LaRC04 criteria, the estimated strength was 360 MPa. The mean FKP in the experiment was 447 MPa (see Table 1); therefore, the Hashin criteria seem to overestimate FKP, and the LaRC04 criteria underestimate it. 4. Determination of design factor One of the problems with composite structures is the excessively conservative design. Large design factors are multiplied with the external load conditions. To avoid conservative designs, the design factors need to be optimized. In this section, we propose reliability-based determination for the design factor based on the FE analysis presented in Section 3. We previously proposed a method to acquire reliability-based design factors in [17]. The method was also adopted in the present study.
Fig. 8. Comparison of experimental and analytical bearing behaviors.
second knee points. A severe reduction in bearing stiffness occurred at the second knee, and the damage extended to inside the specimen. According to the FE analysis results, failures in 0° layers caused the stiffness reductions; this coincides with the prediction based on microscope observation.
4.1. Design parameters
3.3. Prediction for knee point strength
The damage propagation analysis in the previous section had many design variables such as material properties (e.g., strength and elastic constant) and dimension parameters (e.g., clearance between fastener and hole and ply misalignment). Considering all of these variables requires a prohibitive amount of computing cost. To reduce the cost, design variables which significantly affect the knee point strength FKP should be chosen. Some feasibility analyses were performed using the experimental design method and
In Section 2, FKP was regarded as the starting point of apparent nonlinearity on the bearing stress–strain curve. The SCR was developed to detect the knee point. In contrast to noisy experimental data, the FE analysis results seemed little affected by noise. Therefore, we were able to detect the knee point on the analytical rb–eb curves directly. We predicted the knee point strength FKP to
INNER
OUTER
INNER
Before the knee Point
OUTER After the knee point
1400 1200 1000
Transverse-compressive Fiber-tensile Fiber-compressive
800 600 400 200 0 -1
0
1
2
3
4
5
Fig. 9. Damage progression in 0° plies through the knee point (Hashin criterion).
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INNER
OUTER
INNER
OUTER
Before the first knee point
INNER
OUTER
After the first knee point
INNER
Before the second knee point
OUTER After the second knee point
1400 1200
#2: Transverse-compressive #4: Fiber-compressive
1000 800 600
#5: Transverse-compressive
400
#6: Fiber-compressive
200 0 -1
0
1
2
3
4
5
6
Fig. 10. Damage progression in 0° plies through the knee point (LaRC04 criterion).
Fig. 11. Knee point strength FKP estimation by Hashin and LaRC04 criteria.
analysis of variance. Ten variables, modulus in fiber direction EL, that in transverse direction, ET, Poisson’s ratio m (assumed m = mLT = mTR = mLR for computing cost saving), longitudinal shear
modulus GL, compressive strength in longitudinal direction Lc, critical energy release rate in mode I GIc, compressive strength in transverse direction Tc, critical energy release rate in mode II GIIc,
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coefficient for non-linear term in in-plane shear behavior b (contained in SisL implicitly [25]) and fracture angle by transverse compressive load a0 (contained in ST implicitly [25]) were considered. Three variables were screened from the other ones: Lc, Tc, and ET. Other variables were ignored because of negligible effects on the bearing failure behavior and/or their small variability. In both criterion cases, null hypotheses about the three variables were rejected, but the F-statistics of Lc were distinctive; this means that Lc more strongly affected the predictions for FKP than the other parameters. These were called ‘‘design parameters.’’
Table 6 Coefficients and t-statistics for the response surface equation (Eq. (16)) (Unit: [MPa], Hashin Criterion Used).
Coefficients t-Statistics
X ¼ 2ðx lx Þ=d
ð14Þ
where lx is the mean value of parameter x and d is the difference between the maximum and minimum x, respectively. All variables were intended to have a closed interval [1.0, +1.0] through this conversion. Based on the computed results, the form of RS should be nonlinear. Thus, we assumed its form to be quadratic:
F KP ¼ a0 þ a1 Lc þ a2 ET þ a3 T c þ a4 Lc ET þ a5 Lc T c þ a6 ET T c 2 2 2 2 þ a7 L2 c þ a8 ET þ a9 T c þ a10 Lc ET þ a11 Lc T c 2 2 2 þ a12 ET L2 c þ a13 ET T c þ a14 T c Lc þ a15 T c ET
ð15Þ
When the Hashin criteria were used, Eq. (15) was optimized into Eq. (16), and the equation was named F HKP . All the coefficients were determined to maximize the adjusted coefficient of determination R2ad .
R2ad
Their values and t-statistics are listed in Table 6. was more than 99% and seemed to approximate the FE failure analysis well.
F HKP ¼ a0 þ a1 Lc þ a2 ET þ a3 T c þ a4 Lc ET þ a5 Lc T c þ a7 L2 c þ a10 Lc E2 T
ð16Þ
Based on the LaRC04 criteria, Eq. (15) was optimized into Eq. (17) (F LKP ). All the coefficients were determined in the same manner as with the Hashin criteria. Their values and t-statistics are also listed Table 5 Levels for design parameters.
Lv. 1 Lv. 2 Lv. 3
a1
a2
a3
a4
a5
a7
a10
498.2 302.7
123.8 71.9
11.53 6.27
5.05 4.03
1.67 1.79
4.12 4.41
4.54 3.44
2.47 1.52
Table 7 Coefficients and t-statistics for the response surface equation (Eq. (17)) (Unit: [MPa], LaRC04 criterion used).
4.2. Monte Carlo simulation 4.2.1. Response surface method The design parameters were given values called ‘‘levels.’’ First, they were assumed to obey a normal distribution with a mean value equivalent to the experimental value and 5% coefficients of variance (C.V.). The minimum and maximum levels of the parameters were intended to correspond to l 3r and l + 3r, respectively (l = mean value and r = standard deviation). The levels for the three parameters Lc, Tc, and ET are listed in Table 5. They have three levels each, so we were able to generate 33 = 27 combinations; these were substituted into the FE damage analysis presented previously, and we obtained 27 results for each failure criterion. According to the Hashin criteria, the minimum and maximum results were 347.4 and 639.3 MPa, respectively. According to the LaRC04 criteria, these results were 276.1 and 461.9 MPa, respectively. Based on the previous 27 2 results for the failure analysis, we can estimate the response surface (RS) equations that emulate the behavior of the analysis on the variance of the design parameters. For computing accuracy, the normalized design parameter x⁄ was introduced as follows:
a0
Coefficients t-Statistics
a0
a1
a2
a3
a7
a12
363.5 93.5
66.18 17.1
16.50 3.59
1.64 1.96
8.04 2.57
11.63 3.04
in Table 7. The adjusted coefficient of determination R2ad was more than 99%. 2 F LKP ¼ a0 þ a1 Lc þ a2 ET þ a3 T c þ a7 L2 c þ a12 ET Lc
ð17Þ
A comparison of F HKP (Eq. (16)) and F LKP (Eq. (17)) shows that F LKP has fewer terms than F HKP . In particular, coupled terms of Lc* and other variables seem to have been omitted. This is because of the difference in failure criteria. The Hashin criteria contain Lc in explicit form (Eq. (5)), but the LaRC04 criteria do not. Lc is only used as a threshold for LaRC04 #6 (Eq. (13)). Therefore, the LaRC04 criteria are less affected by Lc than the Hashin criteria. Fiber-compressive failure is a significant failure mode for the FKP prediction. As discussed in a later section, this produces a difference between reliability-based design factors. 4.2.2. Monte Carlo simulation Monte Carlo simulation (MCS) was performed to reveal the stochastic properties of the knee point strength of bolted joints in CFRP laminates. The MCS took 20,000 iterations to fully converge. The design parameters were regarded as probabilistic variables assumed in the former section. Thus, the normalized parameters, Lc ; ET , and T c were considered to be probabilistic variables obeying normal distributions with a mean value of 0.0 and standard deviation of 0.33. Fig. 12 shows the MCS results in the form of a histogram and probability density function. The coefficient of variance for the design parameters was 10%. The applied distributions were verified by v2 test. When the Hashin criteria were used (Fig. 12(a)), the results could be applied into a normal distribution (Mean: 497 MPa, SD: 41.3 MPa). On the other hand, the results from the LaRC04 criteria (Fig. 12(b)) were applied into a log-normal distribution (Mean: 364 MPa, SD: 22.7 MPa). The difference was because of the forms of the response surface equations (see Eqs. (16) and (17)). Eq. (16) (Hashin, F HKP ) seemed to have stronger linearity with respect to the design parameters than Eq. (17) (LaRC04, F LKP ). The linear sums of the normal probabilistic variables were also normal probabilistic variables. As previously noted, the Hashin criteria seemed to overestimate FKP, and the LaRC04 criteria underestimated it. According to the experimental results for FKP, its mean value was 447 MPa and the standard deviation was 37.2 MPa (8.3% C.V.). Numerical results suggested standard deviations of 41.3 MPa (8.3% C.V.) and 22.7 MPa (6.2% C.V.) according to the Hashin and LaRC04 criteria; these seem reasonable compared to the experimental values. 4.3. Design factor by probabilistic method
Lc [MPa]
ET [MPa]
Tc [MPa]
1056 1242 1428
6800 8000 9200
124 146 168
4.3.1. Limit stress based on reliability Fig. 13 compares limit stresses—defined as the stress generated when a limit load is applied to the structure[14]—according to the
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Fig. 12. Results of Monte Carlo simulations. (a) Distribution for FKP (Hashin criteria used: 10% C.V.). (b) Distribution for FKP (LaRC04 criteria used: 10% C.V.).
probabilistic and deterministic designs. Fig. 13(a) shows the case with the Hashin criteria, and Fig. 13(b) shows the case with the LaRC04 criteria. The graphs show the probability of failure (POF) and applied bearing stress rb as the vertical and horizontal axes, respectively. The curve is the cumulative density function of FKP found by integration of the probability density functions, and the limit stress by probabilistic design was determined as follows. Three assumptions were made: (1) the applied stress was constant (not a probabilistic variable); (2) the allowable probability of failure was 108; and (3) the margin of safety (MS) was equal to zero. The MS indicates the margin between a load applied to the structure and the allowable load. This value was assumed to be zero:
MS ¼
PA 1 ¼ 0 ) PA ¼ DF PLimit DF PLimit
ð18Þ
where PA is the allowable load for the structure, DF is the design factor, and PLimit is the limit load for the structure. The limit stress is thus introduced as follows [14]:
rA ¼ DF rLimit
ð19Þ
where rA is the allowable stress and rLimit is the limit stress. Based on these assumptions, the stress when the probability of failure is equal to 108 on the cumulative density function was estimated to be 265 MPa (Hashin) and 256 MPa (LaRC04). These values correspond to the probabilistic limit stresses. In deterministic design, the limit stress is determined based on the allowable stress by FE analysis and the fitting factor. FE analysis estimated FKP to be 399 MPa (Hashin) and 312 MPa (LaRC04) when the design
Fig. 13. Comparison between probabilistic and deterministic designs of limit stress for bolted joint in composite laminate (10% C.V.) (a) Hashin criteria: 10% C.V. (b) LaRC04 criteria: 10% C.V.
parameters used A-basis [11] values as their design tolerances. The design tolerance was statistically acquired from the following definition. ‘‘A-basis’’ indicates that at least 99% of the population was expected to equal or exceed the statistically calculated mechanical property value with a confidence of 95%. However, FKP should be regarded as containing a margin derived from the fitting factor. The fitting factor is multiplied with the applied load condition, and the load multiplied by the fitting factor must be lower than the strength of the joint. Therefore, FKP from FE analysis was divided by the fitting factor, which was assumed to be 1.15. Finally, the limit stress for the mechanical joint by the deterministic method was 347 (Hashin) and 272 MPa (LaRC04). The values from the probabilistic method were less than the deterministic values. Not only the failure criteria but also the coefficients of variances affected the results. Fig. 14 compares the probabilistic and deterministic limit stresses when 5% coefficients of variance for the design parameters were assumed. The computed limit stresses seemed to almost correspond to each other. 4.3.2. Reliability-based design factor The results from the probabilistic design method can be converted into the form of a design factor. FKP from FE analysis can be considered to be equal to the product of the probabilistic design factor and the probabilistic limit stress. Therefore, the probabilistic limit stresses under previous conditions can be converted into the fitting factors; these values are presented in Table 8. The conventional design for a mechanical joint requires a fitting factor of more than 1.15. The reliability-based factors are close to the conventional values. However, the value of 1.15 is set for metallic
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stress–strain curve; our research related the nonlinearity, especially in the early steps in the bearing curve, to internal damages (compressive damage in the 0° layers) in the bearing-loaded specimens. This suggestion was supported by microscope observation and FE analysis. The response surface method was used to perform MCS and acquire the stochastic properties of the knee point strength. Designs (limit stresses) by probabilistic and deterministic methods were compared, and the probabilistic design was converted into the form of a fitting factor, and its proper value was proposed. References
Fig. 14. Comparison between probabilistic and deterministic designs of limit stress for bolted joint in composite laminate (5% C.V.) (a) Hashin criteria: 5% C.V. (b) LaRC04 criteria: 5% C.V.
Table 8 Design factors based on failure probability of 108. C.V. (%)
Hashin
LaRC04
10 5
1.51 1.18
1.22 1.12
structures; therefore, actual composite structures have higher values. Thus, the reliability-based design factors in this study seem to be less than the actual values. The design factors vary according to the assumed variances of material properties. The results suggest that composite structures can be refined and made more efficient by evaluation of the variances of the design parameters and factoring them into the design factors. 5. Conclusion A design-oriented strength for bolted joints in CFRP laminates was proposed and analyzed probabilistically. The strength FKP was defined as the stress value at the knee point on the bearing
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