This study presents a Lamb wave-based probabilistic fatigue life prediction for riveted lap joints. First, a brief introduction is given for the experiment of Lamb wave-based damage detection. Three damage sensitive features (correlation coefficient, amplitude change and phase change) are employed to correlate the fatigue crack size with Lamb wave signal. Then the probability of detection (POD) method is used to couple the actual crack size with the model predictions using Lamb wave signal. Considering the uncertainties of the initial crack size and crack growth parameters, Bayesian method and Markov Chain Monte Carlo (MCMC) simulation are applied to obtain the probabilistic fatigue life. In order to verify the reliability of the proposed probabilistic fatigue life prediction procedure, one set of experimental data is used for validation purpose

Dengjiang Wang

Jingjing He

Weifang Zhang

English

Vibroengineering PROCEDIA

  © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533 235 
Lamb wave-based probabilistic fatigue life prediction 
for riveted lap joints 
Dengjiang Wang1, Jingjing He2, Weifang Zhang3 
School of Reliability and System Engineering, Beihang University, Beijing, China 
2Corresponding author 
E-mail: 1wangdengjiang@buaa.edu.cn, 2hejingjing@buaa.edu.cn, 3zhangweifang@buaa.edu.cn 
(Accepted 15 September 2014) 
Abstract. This study presents a Lamb wave-based probabilistic fatigue life prediction for riveted 
lap joints. First, a brief introduction is given for the experiment of Lamb wave-based damage 
detection. Three damage sensitive features (correlation coefficient, amplitude change and phase 
change) are employed to correlate the fatigue crack size with Lamb wave signal. Then the 
probability of detection (POD) method is used to couple the actual crack size with the model 
predictions using Lamb wave signal. Considering the uncertainties of the initial crack size and 
crack growth parameters, Bayesian method and Markov Chain Monte Carlo (MCMC) simulation 
are applied to obtain the probabilistic fatigue life. In order to verify the reliability of the proposed 
probabilistic fatigue life prediction procedure, one set of experimental data is used for validation 
purpose. 
Keywords: riveted lap joints, Lamb wave method, POD model, probabilistic fatigue life. 
 Introduction 
Structural health monitoring (SHM) is of critical importance to provide the safety, reliability 
and affordability for structures [1]. As one of the most common failure modes for aviation 
structures, fatigue damage greatly impacts the long term durability of structures [2]. So fatigue 
diagnosis and prognosis are important parts for structural health monitoring [3]. Lamb wave-based 
non-destructive evaluation (NDE) is widely used in SHM, and the validity of this method is proved 
by [4, 5]. The quality of NDE using Lamb wave-based method depends on many uncertain factors, 
including the sensitivity of inspection instruments, the service condition of target systems [6]. 
Therefore, the reliability and accuracy of the NDE method need to be concerned for an effective 
SHM system. Probability of detection (POD) is one of the common used methods to quantify the 
reliability of NDE method [6, 7], and it evaluates the accuracy of the detection devices and the 
damage quantitative models.  
Riveted lap joints have been increasingly used in aviation structures because of the light weight 
and excellent structural performance [8]. Riveted lap joints are subject to cracks initiation arising 
from stress concentration in the vicinity of the rivet holes. Schindel et al. [9] applied Lamb waves 
using air-coupled transducers to adhesively bonded aluminum plates for damage detection. A 
structural health monitoring strategy is proposed in [10] to detect disbands in a composite lap-joint, 
considering the effect of joint geometry using a within-the-bond approach. 
Among all the studies focus on Lamb wave-based damage detection for lap joints, few have 
been reported on probabilistic fatigue life prediction for the Lamb wave-based damage 
quantitative method. In this study, the POD method is employed to investigate the reliability of 
the Lamb wave-based fatigue damage evaluation, and a probabilistic fatigue life for riveted lap 
joint is obtained. 
 Lamb wave-based fatigue crack quantification 
In this study, piezoelectric (PZT) sensor is employed to perform the fatigue crack detection for 
riveted lap joints under fatigue cyclic loading. The specimens are three rivet rows by five rivets 
wide lap joints and are made of 1.6 mm aircraft grade Al 2024-T3 sheets. Specimens consist of 
two aluminum panels, the top panel has a countersunk hole and the bottom panel has a straight 
LAMB WAVE-BASED PROBABILISTIC FATIGUE LIFE PREDICTION FOR RIVETED LAP JOINTS.  
DENGJIANG WANG, JINGJING HE, WEIFANG ZHANG 
236 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533  
hole. The countersunk hole in the first row is marked as the target area in this study. PZTs are 
used as the actuator to excite guided Lamb wave, as well as the sensor to acquire the wave 
reflection from the boundaries of the fatigue crack at the edge of bolt joints. The pitch-catch 
configuration is employed in this study. The hot spots, which are vulnerable to fatigue crack 
according to existing data, are monitored by travelling optical microscope with a CCD camera 
during the fatigue testing process. Specific experimental information can be referred to [11]. Total 
seven specimens are used in the current investigation. Six of them are made from the same 
manufacture, the other one is from a different manufacture, as shown in Fig. 1(a) (NRC). At each 
of the measurement points shown in Fig. 1(a), fatigue testing experiment is paused and the Lamb 
waves are generated by the actuator and the transmitted waves are collected by the sensors.  
Three crack sensitive features (the correlation coefficient, normalized amplitude, and phase 
change between the damage signal and the baseline signal) are used to quantify the fatigue crack 
size, shown in Eq. (1). The parameters of the model are presented in Table 1. 
ܽ ൌ ܣ ൅ ߙଵݔ ൅ ߙଶݕ ൅ ߙଷݖ ൅ ߙସݔݕ ൅ ߙହݔݖ ൅ ߙ଺ݕݖ ൅ ߙ଻ݔଶ ൅ ߙ଼ݕଶ ൅ ߙଽݖଶ, (1)
where ܽ – rack size, ݔ – correlation coefficient, ݕ – phase change, ݖ – normalized amplitude. 
Table 1. Parameters of the second-order multivariate regression model 
Coefficients Value Coefficients Value 
ܣ 9.3153 ߙହ -9.2745 
ߙଵ -6.8290 ߙ଺ 2.9318 
ߙଶ -8.0771 ߙ଻ 8.1628 
ߙଷ -4.8965 ߙ଼ 4.5682 
ߙସ 9.1995 ߙଽ 4.4369 
In above discussion, a fatigue crack size quantification model based on Lamb wave method is 
introduced to evaluate the crack size. It provides a useful NDE tool for riveted lap joints 
components. Next, the reliability of the Lamb wave-based crack detection model is evaluated by 
the formal of POD. 
 Probability of detection modeling 
The general POD method falls into two categories: hit/miss data and signal response data 
[6, 12]. For hit/miss data, NDE only recorded in terms of whether the flaw is detected or not. But 
for more informative NDE, the signal data can predict the crack size. Thus the NDE data are 
referred to as signal predication, denoted as ොܽ. The POD for signal response data can be derived 
from the correlation of signal predication of crack size (ොܽ) vs. actual crack size (ܽ). 
It has been reported [13] that ln ܽ and ln ොܽ are usually linearly correlated, and can be expressed 
as: 
ln ොܽ ൌ ߙ ൅ ߚlnܽ ൅ ߝ,	 (2)
where ߙ, ߚ are fitting parameters and ߝ is a normal random variable with zero mean and standard 
deviation ߪఌ. 
In order to verify the linear correlation between ln ܽ and ln ොܽ, the log-scale plot using the total 
31 points demonstrated in Fig. 1(a), are shown in Fig. 1(b). A liner regression is used to estimate 
the coefficient and intercept in Eq. 2. ߙ is 0.0611, ߚ is 0.9326, and ߝ~ܰሺ0,0.1237ሻ. The mean 
and 95 % bound prediction results are shown in Fig. 1(b) in solid and dash lines, respectively. 
A pre-defined ොܽ௧௛  is used to represent the detection threshold. The ොܽ௧௛ can be assumed 
according to the measurement noise and physical limits of the measuring devices. It influences 
both the minimum size detected and the probability of false positive detection [7].  
LAMB WAVE-BASED PROBABILISTIC FATIGUE LIFE PREDICTION FOR RIVETED LAP JOINTS.  
DENGJIANG WANG, JINGJING HE, WEIFANG ZHANG 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533 237 
a) Fatigue testing data and life prediction curve 
 
b) Sizing information of actual crack size  
and prediction crack size in log-scale 
Fig. 1. The results of crack size 
If ොܽ exceeds the threshold value of ොܽݐ݄, a flaw is regarded as identified. The probability of 
detection of size ܽ can be detected as follows: 
PODሺܽሻ ൌ Prሺln ොܽ ൐ ln ොܽ௧௛ሻ, (3) 
PODሺܽሻ ൌ Prሺߙ ൅ ߚlnܽ ൅ ߝ ൐ ln ොܽ௧௛ሻ ൌ Φ൬lnܽ െ
ln ොܽ௧௛ െ ߙ
ߚ
ߪఌ
ߚ൘ ൰, (4) 
where Prሺሻ represents the probability of event ሺሻ, Φሺሻ represents standard normal cumulative 
distribution function. Based on the POD model, the probability density function of crack size can 
be obtained. The reliability of the fatigue crack detection method can be classified into two groups. 
First, no crack has been detected by device. It does not necessarily mean the target system is free 
of crack when no crack detected in the NDE results. The inherent uncertainty and the ොܽݐ݄ may 
affect the NDE results. Second, a flaw has been detected. The POD represents the probability of 
detecting a size ොܽ crack when the true crack size is ܽ. Event ܦ defines that a flaw has been detected 
and event ܦഥ  defines that no crack has been detected. For convenience, random variables are 
denoted using capital letters (e.g., ܣ) and corresponding values are denoted using lower case 
letters (e.g., ܽ). 
 Indication is found in NDE data 
The detected crack size is denoted as ܽ′. The distribution of the actual crack size can be 
expressed as the following equation [14]: 
஺݂|஽ሺܽሻ ൌ
1
ܽሺߪఌ ߚ⁄ ሻ
߶ ቆlnܽ െ lnܽ
ᇱ െ ߙ
ߚ
ߪఌ
ߚ൘ ቇ, (5) 
where ܣ represents the random variable of the actual crack size, and ߶ሺሻ is the standard normal 
probability density function (PDF). 
 No indication is found in NDE data 
When there is no crack indication is found in NDE results, the PDF of the actual crack size 
can be expressed as Eq. (6) [14]: 
஺݂|஽ഥሺܽሻ ൌ
ሾ1 െ PODሺܽሻሿ ஺݂ሺܽሻ
׬ ሾ1 െ PODሺܽሻሿ ஺݂ሺܽሻ݀ܽ
ஶ
଴
. (6) 
0 0.5 1 1.5 20
0.5
1
1.5
2
Logarithmic experimental crack size (mm)
Lo
ga
rit
hm
ic 
pr
ed
ict
ion
 cr
ac
k s
ize
 (m
m)
 
 
Training data
Median prediction
95% CI
LAMB WAVE-BASED PROBABILISTIC FATIGUE LIFE PREDICTION FOR RIVETED LAP JOINTS.  
DENGJIANG WANG, JINGJING HE, WEIFANG ZHANG 
238 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533  
 Probabilistic fatigue life assessments for lap joint 
 Uncertainties of fatigue growth model parameters 
There are many sources of uncertainties for fatigue life predication. One of the critical factors 
is initial crack size. The initial crack size estimation based on NDE data has been discussed above. 
The uncertainties from material properties and geometries are also need to be included. In general, 
the material uncertainties affect the fatigue life prediction through fatigue model parameters. The 
classic Paris’ equation (shown in Eq. (7)) is employed here for fatigue life predication. The fatigue 
testing data (shown in Fig. 1(a)) for the lap joint specimens are used to preform Bayesian 
parameter estimation with Markov Chain Monte Carlo (MCMC) method: 
dܽ/dܰ ൌ ܥሺΔܭሻ௠,	 (7)
where ܽ is the crack size, ܰ is the number of cyclic loads, Δܭ ൌ Δߪ√ߨܻܽ is the stress intensity 
factor range during one loading cycle, Δߪ  is the applied stress amplitude, ܻ  is the geometric 
correction factor, and ܥ and ݉ are model parameters. 
The joint distribution of ሺln ܥ ,݉ሻ is estimated using 200,000 MCMC samples, considering 
the two parameters follow a multivariate normal distribution (MVN). The mean vector is 
ߤሺlnܥ,݉ሻ ൌ ሾെ23.1670, 2.6214ሿ and the covariance matrix is ∑ 	ሺ୪୬஼,௠ሻ ൌ ቀ
10.7557 െ1.8394
െ1.8394 0.3133 ቁ. 
 Probability fatigue life prediction 
The critical stress intensity factor of the lap joint material is ܭܫܿ ൌ 24.5	MPa√mm, and the 
critical crack size is obtained as ܽܿ ൌ 19.036	mm. Using the direct integration of Eq. (7), the 
probability fatigue life can be expressed as: 
ܰ ൌ න 1
Δߪ√ߨܻܽ
௔೎
௔బ
dܽ,	 (8)
Δܭ ൌ Δߪ√ߨܻܽ,	 (9)
where ܽ0 is the initial crack size, Δߪ is the applied stress amplitude, ܻ is the geometric correction 
factor and equal to 1 for model I case. 
 The probability of fatigue life with no flaw identified 
Based on the fatigue testing data, the minimum detected crack size is obtained as 1.42 mm. 
Thus 1.42 mm is considered as detection threshold (i.e., ොܽݐ݄ ൌ  1.42 mm). Using numerical 
quadrature to evaluate the normalizing constant (denominator) of Eq. (6), the actual flaw size 
distribution is given by: 
݂ܣ|ܦഥሺܽሻ ൌ 0.727 ൤1 െ Φ൬
lnܽ െ 0.3105
0.1326 ൰൨. (10)
The PDF of the actual crack size with no indication is shown in Fig. 2. The blue line represents 
the actual crack size distribution, given that no crack is identified from the NDE data and the 
detection threshold is 1.42 mm. 
Using 100,000 times of MC simulations, the probability of fatigue life with no indication 
(detection threshold is 1.42 mm) is shown in Fig. 3(a). 
LAMB WAVE-BASED PROBABILISTIC FATIGUE LIFE PREDICTION FOR RIVETED LAP JOINTS.  
DENGJIANG WANG, JINGJING HE, WEIFANG ZHANG 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533 239 
 
Fig. 2. The probability density functions of the actual crack size. 
 The probability of fatigue life with a flaw identified 
Given the fact that a crack with a size of 1.50 mm is reported, the PDF of the actual crack size 
(using Eq. (5)) is: 
݂ܣ|ܦሺܽሻ ൌ
1
0.1326ܽ߶ ൬lnܽ െ
ln1.50 െ 0.0611
0.9326 0.1326ൗ ൰. (11) 
The PDF of the actual crack size with the flaw indication is shown in Fig. 2. With 100,000 MC 
simulations, the probability of fatigue life with an identified flaw (crack size is 1.50 mm) is shown 
in Fig. 3(b). 
 
a) The probability of fatigue life with no flaw 
identified: threshold detection is 1.42 mm 
 
b) the probability of fatigue life  
with a flaw identified (1.50 mm) 
Fig. 3. Probability of fatigue life for the lap joints 
 
Fig. 4. The probability of fatigue life with flaw identified (1.61 mm) 
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
the actual crack size (mm)
Pr
ob
ab
ilit
y d
en
sit
y f
un
cti
on
 
 
The threshold is
 1.42mm
a 1.50mm crack is
first detected
0 1 2 3 4
x 105
0
0.02
0.04
0.06
0.08
0.1
0.12
Fatigue life
Pr
ob
ab
ilit
y
2 2.5 3 3.5 4
x 104
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Fatigue life
Pr
ob
ab
ilit
y
1 1.5 2 2.5 3
x 104
0
100
200
300
400
500
600
700
800
Fatigue life
Pr
ob
ab
ilit
y
LAMB WAVE-BASED PROBABILISTIC FATIGUE LIFE PREDICTION FOR RIVETED LAP JOINTS.  
DENGJIANG WANG, JINGJING HE, WEIFANG ZHANG 
240 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533  
 Validation with experimental data 
In order to verify the reliability of the proposed probabilistic fatigue life prediction procedure, 
T5 specimen is used for validation. The minimal detected crack is reported with a size of 1.61 mm 
at ܰ ൌ 14845 cycles. The specimen failed at a crack size of 7.24 mm. Using 1.61 mm as the 
reported crack size and 7.24 mm as the critical crack size, the probabilistic fatigue life results are 
obtained using 100,000 MCMC simulations, as shown in Fig. 4. The experiment result 
(14845 cycles) is within the 95 % confidence interval (12444, 22770), and using the lower bound 
of 12444 can provide a small safety margin for decision-making. 
 Conclusions 
In this study, a new methodology for the probabilistic fatigue life prediction of riveted lap 
joints is proposed. A multi-feature integration crack detection model is used to correlate the Lamb 
wave signal to the actual crack size. A log-linear POD modeling for Lamb wave-based NDE is 
investigated based on the multi-feature crack detection model. The PDFs of the actual initial crack 
size is obtained using the log-linear POD model. MCMC method is used to perform the parameter 
estimation for the initial crack size and crack growth model parameters. The probabilistic fatigue 
life for riveted lap joint is obtained using Monte Carlo simulations. Additionally, one set of 
experimental data is used to validate the reliability and accuracy of the proposed probabilistic 
fatigue life prediction. 
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Lamb wave-based probabilistic fatigue life prediction for riveted lap joints

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Lamb wave-based probabilistic fatigue life prediction for riveted lap joints

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