Abstract
In this paper, we propose a stochastic model to describe over time the evolution of stress in a bolted mechanical structure depending on different thicknesses of a joint elastic piece. First, the studied structure and the experiment numerical simulation are presented. Next, we validate statistically our proposed stochastic model, and we use the maximum likelihood estimation method based on Euler–Maruyama scheme to estimate the parameters of this model. Thereafter, we use the estimated model to compare the stresses, the peak times, and extinction times for different thicknesses of the elastic piece. Some numerical simulations are carried out to illustrate different results.
1. Introduction
The bolted structures are widely used in automotive and aeronautical applications, and they ensure very good rigidity of the assembly and high level of security. The dynamic modeling of bolted structures is the subject of numerous works in the industrial as well as in the academic community, and it represents a real challenge for mechanical engineers. Several studies [1–3] have been carried out to model this kind of structure, with taking into account the influence of diverse solicitations such as creep, pressure, thermal charging, vibration, and fatigue. In this framework, putting an elastic joint piece between two plates of the assembly allows to obtain a great stability of the structure. The purpose of this technique is to absorb the distribution of stresses causing by the existing of significant vibration transfer and von Mises stress, from one plate to another.
Our original contribution in this paper consists to model the influence of the existence of an elastic piece on the dynamic behavior bolted mechanical structure which is under the effect of von Mises stress. We start out in Section 1 with a description of the studied bolted structure; then, we use in first time a finite element (EF) analysis approach to investigate the parameters that may affect the stress concentration in a bolted assembly, depending of different thicknesses of the elastic piece. The EF numerical simulation by ABAQUS software is described, and the results of the simulations are presented (simulation inputs and obtained data) in Section 2.
Thereafter, this data are exploited to evaluate the correlation between the stress applied and the thickness of the joint elastic piece. In other terms, we are modeling stochastically the stress variation over time of a bolted structure in function of the thickness of a joint elastic piece. This correlation is made by a Pearson test.
Therefore, a stochastic model is built, and its parameters are estimated; this new model allows predicting the point of stress stability (stress peak) of behavior of bolted structure studied, its time (stress peak time) and its extinction time. First step is to verify the stress distribution normality by a graphical test, which is carried out by the Q-Q plot (Quantile-Quantile plot) model; this tool is used to determine the best fit for the duration of phonemes. The q-q plot shows the relationship between the quantiles of expected distribution and actual data. Also, a Shapiro–Wilk test is made to demonstrate the distribution normality by the analytical method.
Section 3 aims to validate the effectiveness of the proposed stochastic model and to estimate its parameter by the Euler-maximum likelihood estimation method. Section 4 is dedicated to predict the stress peak and the stress extinction time and to discuss the results of our modeling. In Section 5, new simulations are carried out with a steel piece with different thicknesses, in order to compare their results with the previous results obtained for the bolted structure with the joint elastic piece. We finish by a conclusion and some perspectives.
1.1. State of Art
In the literature, several mathematical methods have been handled in order to ensure the validity of bolted structures. The finite element (EF) analysis has been used to examine the bolt hole clearance effect on the mechanical behavior of a bolted structure [4]. Also, Hashin’s failure criteria have been used to predict the failure onset load (see [5, 6]). These studies applied principally analytical methods supported with some included experimental results and lead to draw several conclusions to model more complex structures.
Also, several stochastic models have remained an effective tool to analyze the behaviors of mechanical structures by taking into consideration the randomness of mechanical properties. These models introduce uncertainties in the parameters of deterministic models by supposing that they are subject to environmental fluctuations, which gives more realism to the results.
Constructing and studying deterministic and stochastic models belong to the most beneficial methods to estimate the relation between the input parameters of the structure finite element model and the response parameters of interest. In this context, Mccarthy and Gray [7] proposed and analyzed a deterministic model for predicting the distribution of loads in multibolt composite joints [8]. In other investigation, Lacour et al. [9] have modeled the von Mises stress, stiffness, and displacements by a nonlinear stochastic model at each degree of freedom.
In order to formulate a stochastic finite element method for nonlinear material models, the same authors have applied a discrete approach to develop a constitutive algorithm which can be implemented on a global level of the context 3D nonlinear stochastic finite element method [10]. In this work, our proposed modeling is done by the continuous stochastic model, and it should be noticed that it has not been treated before in the literature.
2. Experiment Numerical Simulation and Data
2.1. Description of the Studied Structure
For aeronautic and automotive applications, bolted assemblies are subjected to traction, compression, torsional torque, and thermal and centrifugal forces. As presented in Figure 1, the studied bolted structure is submitted to compressive stress, and an elastic piece is put between the two steel plates in the assembly area. Figure 1 indicates all the ratings used. The parameters related to the substrate are the thickness of the elastic piece, the width, and the length of the two plates, as well as the diameter of the hole from the middle to the contact zone.
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2.2. Compression Test Simulation
First, we aim to simplify the meshing of the finite element (FE) model of the studied structure, by substituting its geometry with elements having an equivalent behavior. The simulations using ABAQUS software are carried out. ABAQUS is an excellent software tool which can incorporate the nonlinearity of materials, geometry, asticity (strain hardening), large displacement, contact problem, etc. The geometry of the structure was created by using the modules Parts and Assembly in ABAQUS software (see Figure 2).
The material properties for the different pieces of the structure have been chosen according to the type of material. For the materials of construction, the two plates and the bolt are made of steel, while the elastic piece is made of rubber. The material properties of the different pieces of the structure are detailed in Table 1:
In this simulation, compression loads of 50 N/mm2 was applied to the side face of the upper plate (see Figure 2), and we modified the thickness of the elastic piece to study the dynamic response of the structure and then, to model the von Mises constraint distribution in the lower plate of the structure, which depends on time and thickness.
All the simulations below are performed using an explicit dynamic step. The contact properties between the parts of the structure are defined. During this meshing, we obtain a three-dimensional continuum of 8 inclined brick elements. Finally, we can run the simulation and extract the results.
2.3. Result Finite Element Simulation
Finite element (FE) analysis is a relatively inexpensive and fast alternative to physical experiments. Reliable test data are essential to calibrate an FE model. If the validity of FE analysis is ensured, it is possible to model the dynamic response of the structure with the number of parameters.
The elastic piece thickness is modified from 0 mm to 2 mm and for each thickness a numerical simulation is made. For each simulation step, we apply the same conditions in order to investigate thickness effect in the distribution of the vonMises stress applied in the lower plate of the structure. Figure 3 shows the simulation results for only 7 types of thicknesses, in order to study the stresses distribution on the lower plate in function of the thickness of the elastic part.
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The results of the simulations show that the stresses are concentrated in the contact zone, between the lower plate and the elastic piece. In the case of the structure without elastic piece, the stresses are distributed around the holes. With a thickness of 0 mm, it can be seen that the stresses are distributed over the entire lower plate and that its distribution reaches its maximum in the periphery of the hole.
For each simulation, an element of the lower plate is chosen in the contact area where the stress distribution is maximum. In order to efficiently observe the influence of the thickness change on the stress evolution and to properly interpret the results, we make a representation of this stress as function of time. The simulation results for 7 thicknesses are presented in Figure 4.
It can be seen that the stress in the bolted structure with elastic piece thickness 0, 0.1, 0.7, and 1 is significantly higher compared to the stress in the structure bolted with elastic piece thickness 0.4 and 0.9. Accordingly, the simulation result reveals that there exists any correlation between the stress evolution and the elastic piece thickness. In order to confirm the last hypothesis, a Pearson test is carried out. Table 2 summarizes the results of this test.
As can be see from the results above the value of the test is 0.7794, which is bigger than the significance level . We can conclude that the stress and the joint elastic piece thickness are not correlated. Officially, the thickness does not affect the stress.
In the following section, a stochastic model is built, and its parameters are estimated.
3. Model and Parameter Estimation
3.1. Stochastic Model of Modelisation
The QQ plot (or quantile-quantile plot) establishes the correlation between a given sample and the normal distribution. A 45-degree reference line is also drawn. In a QQ plot, each observation is plotted as a single point. If the data are normal, the points should form a straight line. The following figure gives the QQ plot linked to the stress sample of each elastic piece thickness.
We can remark from Figure 5 that all points lie approximately along this reference line, and we can assume the normality of the stress sample linked the given elastic piece thicknesses. Visual inspection, as described in previously, is generally unreliable. A significance test comparing the sample distribution to a normal distribution can be used to determine whether or not the data shows a significant deviation from the normal distribution. There are several methods for assessing normality, including the Kolmogorov–Smirnov (K-S) normality test and the Shapiro–Wilk test. The Shapiro–Wilk test is widely recommended for normality testing and provides better power than K-S test. It is based on the correlation between the data and the corresponding normal scores (see Ghasemi and Zahediasl [11]). The following table summarizes the results of the Shapiro–Wilks test for all the samples of stresses with different elastic piece thicknesses.
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According to Table 3, for each elastic piece thickness, the value indicates that the corresponding stress distribution is not significantly different from the normal distribution. In other words, we can assume normality of all the distribution. Thus, if is the one-dimensional variable that represents the stress density linked to a piece elastic thickness per second, sowhere , and are constant characteristics. By deriing along time, we obtainand we suppose that the parameter is subject to random fluctuations. We replace by in equation (2), where is a standard one-dimensional Brownian motions and is the intensity of the perturbation. We obtain the following stochastic model:
3.2. Parameter Estimation
Let a time interval and be a subdivision of this interval. We suppose that the step for all . The numerical approximation of model (3) by Euler–Maruyama scheme giveswhere . We suppose that , so the variablesare independent and identically distributed (iid) and that all follows the law . Thus, the density function of any iswhere . The Euler-ML estimator of is concerned to find the parameter vector that maximizes the log-likelihood function:
The MLEs , and for the parameters , and verifyi.e.,
By resolving this problem, we found the estimators , and of , and :where
For , we have
Thus, the stochastic modelcan describe the evolution of von Mises stress in a time interval .
In the remainder of this paper, it is assumed that the mechanical action is suppressed when the stress reaches the point of stability (peak of the stress).
4. Simulations and Discussion
Depending on experimental sample of each thickness, we calculate the linked estimators , , and using formulas (10), (11), and (13). Table 4 gathers the results.
As example, the estimating model for the elastic piece thickness 0.2 mm is
In the following simulations and table, the peak stress (stress point of stability) and its peak time are stochastically estimated using model (14). We recall that when the stress reaches the point of stability, it is assumed that the mechanical action is suppressed. The stress extinction time is also estimated.
As indicated in Figure 6 and Table 5, we can analyze these results in two levels; first, the stress peak time and the extinction time are not correlated with the elastic piece thickness, but the variance of their evolution are very small. In other words, the elastic piece thickness affects slowly the stress peak time and the stress extinction time, although there exist no correlation between the elastic piece thickness and the other two variables. In the second level, the stress peak is also not correlated with the elastic piece thickness; on top of that the variance of the evolution of this variable with change of the elastic piece thickness is very large.
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In general, the mechanical loads applied to the upper plate are transferred to the lower plate and this transfer is reflected in the concentration of stresses on the lower plate. According to the result of Figure 3, the stress is well distributed on all the lower plate in the case of absence of the elastic piece; however, its presence concentrates the stress in the lower plate in the contact area around the hole. Thus, the elastic part allows to concentrate the stresses in the contact area or in the assembly area, where there is the bolt and its action. This helps to reduce the damage of the structure because the assembly area is one of the hardest part in this structure.
On the contrary, as deduced in the correlation and stochastic model results, the change in the thickness of the elastic piece does not influence the evolution of stress in the case of the presence of mechanical actions or in the case of their absence. The noninfluence of the thickness on the stress evolution as a function of time in a finite element of the structure is caused perhaps by the material properties of this elastic piece.
The paper of Saxena et al. [12] allows us to consider the last hypothesis. In fact, their paper shows that the stress evolution depends on the piezoelectric layer length and the thickness variation of the plate, which has the same material properties of the global structure and affects the stress evolution.
4.1. Dynamic Behavior of Bolted Structure with Steel Piece
In order to verify this hypothesis, we carry out the same simulations on the ABAQUS Software, but this time, we replace the elastic piece by a steel piece which has the same material properties of the structure. The results of this numerical simulation as well as the correlation test are presented in Figure 7 and Table 6.
The stress evolution is presented as function of time in a lower plate element and the change of the steel piece are strongly correlated. This result is confirmed by the Pearson correlation test in Table 6 (the value is ).
This result is consistent with the hypothesis that we have previously proposed. Therefore, the noncorrelation between the stress evolution and the change in the thickness of the elastic piece is mainly caused by the material properties of the latter.
5. Conclusion
In this paper, an experiment numerical simulation has been conducted to study the von Mises stress evolution into a bolted structure in presence of an elastic piece. The experiment changes the thickness of the elastic piece to deduce its effect in the evolution of the stress. The results of this investigation are various. First, the presence of the elastic piece makes possible to concentrate the stress in the assembly area in the lower plate. The second result also showed that there is no correlation between the stress evolution and the change of the thickness. This uncorrelation is caused by the difference of material properties between the elastic piece and the other pieces of the structure.
A stochastic model has been built from the graph of the stress evolution as well as by a normality test, and its pentameters have been estimated by the Euler-Maximum Likelihood estimation method. The model has been used to predict the stress peak, the stress peak time, and the stress extinction time into the bolted structure. The proposed stochastic model remains a relevant model which gives good predictions of the last three parameters by changing the thickness of the elastic piece despite the fact that there is no correlation between the change in thickness and the stress evolution.
As a perspective, we can consider this work as a basis for several future studies, for example,(i)Build a stochastic model or a time series that predicts the evolution of the stress in function of the time and the thickness of the elastic piece(ii)Study the mechanical reliability of bolted structures with and without elastic part(iii)Search the optimal joint configuration that ensures the stability and reliability of the bolted structure
Data Availability
The datasets used to support the results of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.