The cantilever beam is the main part of cantilevered low-pressure die casting machines. It bears the gravity of other structures as well as the mold filling pressure, dwell pressure, and ejection force. The failure of the cantilever beam can cause serious consequences, and it is difficult to replace or repair. Many research works have studied ways to avoid the failure of the cantilever beam[1-7]. The safety factor method was adopted in the traditional design to set the safe work limit and ensure that the mechanical parts do not fail. For large parts, the safety factor chosen by experience is not accurate. However, the safety factor, strength, and stress are all determined by the design method. Although the safety factor of some parts is greater than 1, a few parts still fail within the prescribed period of use. This is because the strength, stress, and size are random variables in the working state of the parts, and they have a certain discrete type. The reliability analysis method considers factors such as the material, load, and size of the parts as random variables and calculates the probability of structural stress greater than the allowable strength of materials by probability theory [8]. The traditional methods include the Monte Carlo method and response surface method. The response surface method is generally applied to structures with less than 20 input parameters, and the number of cycles depends on the number of input variables. The Monte Carlo method is not affected by the complexity of the structure. When analyzing the complex parts, the sample size increases and the computation increases with the increase in input variables [9-10].

Based on the reliability theory, this paper presents a hybrid stochastic finite element method to analyze the reliability of parts. Combined with the ANSYS/PDS module, firstly, the sensitivity analysis is conducted to examine the impact of inputs on the influencing factors via the Monte Carlo method. According to the analysis results, the structural parameters are optimized. Then, the response surface of the cantilever beam is fitted by the response surface method, and the reliability of the cantilever beam is calculated. This paper firstly introduces the basic theory of the reliability analysis of parts, compares the different finite element analysis methods, and briefly describes the idea of hybrid stochastic finite element simulation. The reliability of the cantilever beam is then analyzed based on ANSYS software.

The definition of reliability is the ability of the product to complete the specified function within the specified conditions and time. The factors that affect the failure of mechanical parts can be divided into two types: strength Y and stress X. Strength refers to the ability of the material itself to resist stress. Stress is the factor that causes the failure of parts, such as force, bending moment, and torque. When $~X>Y,$ the part is considered to be failed; when $~X<Y,$ the part is reliable. In the working environment, the strength and stress of parts will be discrete due to the objective factors such as environment and external forces, which will become a random variable complying with some distributions. The stress can be expressed as $~f(x)$, and the intensity is expressedas $g(y)$. If the two are compared in the same coordinate system, the area where the cross interference occurs may fail [11-12]. The probability of failure is calculated according to the interference situation:

Where $R$ is called reliability. $~X$ and $~Y$ are two independent random variables. $~Z=Y-X,$ and the joint probability density function of $Z$ is $~f(Z)=f(X)g(Y)$. The probability that the strength $Y$ is greater than the stress $~X$ is:

Therefore, the integral region of (1) is obtained, as shown in Figure 1.

If the stress $X$ and the strength $Y$ follow the normal distribution, the mean and standard deviation are $\bar{X}$, $\bar{Y}$, and ${{\sigma }_{x}}$, ${{\sigma }_{y}}$. $Z=Y-X$ is also a random variable that complies with normal distribution; the average value is ${{\sigma }_{Z}}=\sqrt{\sigma _{X}^{2}+\sigma _{Y}^{2}}$, and the probability density function is:

The distribution diagram is shown in Figure 2. The shaded part is the failure probability of the structure, and the non-shaded part is the reliability of the structure.

Probabilistic design technique is used to evaluate the influence of uncertainty of input parameters on system output behavior, and its characteristics and can be used to analyze the possibility of “failure” of parts. The input parameters include uncertain factors such as geometry size, error, material, load, etc. Response parameters include temperature, stress, displacement, etc. The PDS module of ANSYS is based on probabilistic design technique and combines finite element analysis with probabilistic design technology to evaluate the influence of random parameters on model analysis under random conditions.

The mainstep for reliability analys is of parts using the ANSYS / PDS module are as follows:

$\cdot$ Create an analysis file. The analysis process of this file includes: defining parameters, creating models, meshing, loading and solving, and post-processing. In probability design, the whole process must be applied to ANSYS parametric design language. Complex models can be modeled in other three-dimensional softwares, imported into ANSYS using the software’s data interface, and transformed into a model written in ANSYS Parametric Design Language(APDL) via the ANSYS/Workbench software.

$\cdot$ Establish a probabilistic finite element analysis database and all parameters. Enter the PDS module and specify analysis files.

$\cdot$ Define random output parameters and random output parameters.

$\cdot$ Select the tools or methods of reliability analysis. ANSYS provides three computing methods: Monte Carlo method, response surface method, and user-defined method.

$\cdot$ Perform probabilistic design analysis.If an error occurs in the loop, the loop analysis needs to be restarted. In order to improve the efficiency of computing, a small number of cycles are carried out first, and a large number of cycles can be carried out after there are no errors.

$\cdot$ The cycle ends. Carry out post-processing, such as displaying the histogram of the value distribution of random parameters, namely RP, viewing the failure probability distribution function of RP, calculating the probability when RP is less than a certain value, etc.

The data flow of the above analysis process is as follows:

The stochastic finite element method, also called the probabilistic finite element method, is an analytical method that combines the traditional finite element method with the stochastic analysis theory [13]. In the field of structural engineering, a large number of uncertain factors cause the structural size, material, and load to become variables. In the process of random finite element analysis, these variables are random variables that comply with a certain probability distribution. Based on the finite element method, the failure probability of the structure under given conditions is calculated by combining probability theory with mathematical statistics theory and strength theory.

The finite element analysis generally consists of three processes: grid division, grid analysis, and structural analysis. The factors that affect the stiffness of part structure are geometrical dimension, material parameter, and load vector. The geometrical dimension is represented by matrix $\left[ A \right]$, the material parameter is represented by matrix $\left[ B \right]$, and the load vector is represented by the matrix $~\left[ C \right]~$. The strength of the part can be expressed as: $\left[ \text{N} \right]=\int {{\left[ A \right]}^{T}}\left[ B \right]\left[ A \right]$. The stress of the part can be expressed as $\left[ F \right]=\left[ \text{N} \right]\left[ q \right]$, where $\left[ q \right]$ is the displacement of element nodes [10]. The random finite element method, which combines the Monte Carlo sampling simulation, treats the parameters that affect the parts as random variables. For convenience of calculation, the random variable can be decomposed into the sum of its mean and wave values:

When using the stochastic finite element method to calculate reliability, the Monte Carlo method and response surface method are usually combined, and they are called the Monte Carlo stochastic finite element method and the response surface stochastic finite element method.

As a common numerical simulation technique of probability analysis, the Monte Carlo method can accurately simulate the behavioral characteristics in practical situations. It is not limited by the structural state function, so it is widely used in engineering. The principle of the Monte Carlo method is a random simulation test method based on the theorem of large numbers. Through the random simulation test using limited test data and probability distribution model of data, a large number of simulation samples of test data in actual engineering are obtained. The random simulation test of random simulation experiments include direct sampling and Latin hypercube sampling.

Using the Monte Carlo simulation requires the types of the probability distribution of the random variable as well as the expectation and variance. The size of the random variables generally includes the size of the structure, modulus of elasticity, density, yield strength, equivalent force, and equivalent torque [14-15]. There are $~M$ dimensional random vectors $\mathbf{K}={{\left\{ {{K}_{1}},{{K}_{2}},{{K}_{3}},\cdots ,{{K}_{M}} \right\}}^{T}}$, each of which the probability density function of random vector is setas $f\left( {{K}_{i}} \right)$. The random number generating method is used to generate a large number of input variables of the random number ${{R}_{i}}$, and one has ${{R}_{i}}=\underset{-\infty }{\overset{{{X}_{i}}}{\mathop \int }}\,f\left( {{K}_{i}} \right)\text{d}{{K}_{i}}$. Then, the random number is sampled $~X$ times, and one can obtain $~X$ random numbers, represented by ${{R}_{1}},\text{ }{{R}_{2}}\text{, }{{R}_{3}},\cdots \text{, }{{R}_{x}}$. Then, combining Equation(3), each random number is calculated, and the $~X$ discrete variable is obtained. Finally, the probability statistics of discrete variables are calculated based on the large number theorem.

The sampling method adopts the Latin hypercube sampling method [16]. Compared with the Monte Carlo direct sampling method, the Latin hypercube sampling method can avoid the random sampling of the unity and clustering, sampling area stratified, and uniform sampling. This can reduce the sampling frequency and improve the computational efficiency under the premise of obtaining the same sampling precision. The principle is shown in Figure 4.

For reliability analysis of large structures, the Monte Carlo stochastic finite element method will cause a larger number of random variables. The larger the sample size, the more accurate the results, but the higher the computational cost. When the sample capacity increases, sensitivity analysis can be adopted to obtain the sensitivity coefficient of the influence of each random variable on the structural reliability. Reduce the computational cost by ignoring some random variables that have less impact.

Reliability sensitivity is the ratio of failure probability changes caused by changes in basic variable distribution parameters, expressed by partial derivatives of failure probability ${{P}_{f}}$ to random variable distribution parameters ${{\theta }_{x}}.$ The formula is $\frac{\partial {{P}_{f}}}{\partial {{\theta }_{x}}}$. Sensitivity can basically reflect the influence of random variable distribution parameters on failure probability. Thegreater the sensitivity coefficient, the greater the influence on reliability. Sensitivity includes global sensitivity and local sensitivity. The global sensitivity reflects the functional relationship between the partial derivative $\frac{\partial {{P}_{f}}}{\partial {{\theta }_{x}}}$ and the basic variable distribution parameter ${{\theta }_{x}},$ while the local variables reflect the function values of partial derivative $\frac{\partial {{P}_{f}}}{\partial {{\theta }_{x}}}$ in a given ${{\theta }_{{{x}_{i}}}}$[17]. The most commonly used sensitivity analysis method is global sensitivity. Reliability sensitivity analysis can be expressed as:

Where ${{P}_{f}}$ is the failure probability of reliability sensitivity and $\theta _{{{x}_{i}}}^{(k)}$ is the distribution parameter of random variable ${{x}_{i}}$, where ($i=1,2,3,\cdots ,n$;$k=1,2,3,\cdots ,{{m}_{i}}$, and ${{m}_{i}}$ is the total number of distribution parameters of the random variable ${{x}_{i}}$). The Monte Carlo method can be used to solve the sensitivity after transforming the above Equation [18]. Based on the ANSYS / PDS module, this article uses the Latin hypercube sampling method to analyze the sensitivity of the structure.

When dealing with more complex structures, the limit state functions are mostly implicit, and the functional relation between input variables and output variables is not known. The response surface method is a method based on statistical method, which reasonably selects sampling points and iterative methods and finds an approximate function between input variables and output variables to replace implicit functions [19]. For a structure, different methods of fitting may yield innumerable approximate functions. The response surface is usually expressed by a quadratic polynomial:

Where ${{\beta }_{0}},\text{ }{{\beta }_{i}},\text{ }{{\beta }_{ii}}$ are unknown coefficients to be determined by least squares. ${{x}_{1}},\text{ }{{x}_{2}},\text{ }{{x}_{3}},\cdots ,\text{ }{{x}_{i}}$ are random variables. ${Z}'$ is an approximation function, which is used instead of the implicit function $Z$, and ${g}'(x)$ is used to replace the real limit state function $g(x)$. As can be seen from Equation (4), the calculation quantity is related to the number of random variables, where $g(x)$ is affected by random parameters, including material, structural dimensions, and external loads. $g(x)=0$ is the boundary of the reliability constraint. If $~g(x)>0$, the structure is secure. If $g(x)<0$, the structure fails. Using ${{f}_{x}}(x)$ to represent the joint probability density of random variables, the probability of structural failure can be approximated as:

By Equation(5), the reliability is $R=1-P$.

The process of the response surface stochastic finite element method based on the ANSYS platform is basically the same as the Monte Carlo stochastic finite element method. However, the difference of the process is that the response surface of the structure is first fitted before Monte Carlo sampling simulation, then the fitted response surface is used to replace the finite element model of the structure for sampling simulation, and finally the simulation results are obtained. Therefore, the response surface obtained by fitting will affect the results of the next sampling simulation, which will eventually affect the sampling results. The specific flow of response surface stochastic finite element method is shown in Figure 5.

The hybrid stochastic finite element analysis method combined with the sensitivity analysis method of the Monte Carlo and the response surface method can analyze the reliability of the parts. First, the Monte Carlo method is used to sample the influence parameters of the parts, and the sensitivity analysis of each parameter is obtained. Then, the parameters are optimized according to the sensitivity results. Then, the response surface method is used to analyze the optimized parameters, the response surface of the parts is obtained, and the reliability of the part is computed [20]. The reliability analysis method of the component is the same as the Monte Carlo method and the response surface method, and the analysis process is shown in Figure 6.

3D modeling of the cantilever beam is carried out by UG, and the 3D model is shown in Figure 7. The cantilever beam has a fixed end and a free end. The fixed end is bolted to the rotatable column and other parts are installed on the free end. A working cycle of low pressure cantilever casting machine includes casting filling, pressure forming, and stripping. During the casting filling process, the liquid alloy is pressed into the cavity by the riser when the pressure is applied. The filling pressure ${{N}_{1}}$ is about 3600kN. The filling force is transferred to the cantilever beam through the other structure, so that the cantilever beam is subjected to a vertical upward bending moment. During the solidification process of low pressure casting, the filling pressure remained stable until the liquid alloy was cooled and solidified, the pressure was unloaded, and the cantilever beam was continuously affected by the bending moment. In the drawing of the patterns process, the ejection force is about 352kN. In addition, the free end of the cantilever beam carries the gravity of other institutions, about 20kN.

In this paper, the reliability of the cantilever beam is analyzed based on the ANSYS/PDS module. In order to improve the efficiency of the analysis, the structure of the cantilever beam is simplified, and the size parameters of the simplified structure are shown in Figure 8.

Then the unit is set, unit and load is solved. SOLID186 is adopted to divide the unit, and the structure of the grid is divided as shown in Figure 9(a). In order to ensure the grid quality, the sharp corners of the structure are refined, as shown in Figure 9(b).

The material of the cantilever beam is Q235, and the material characteristics are shown in Table 1. The load spectrum of the cantilever beam during the filling period and the drawing of patterns process is shown in Figure 10 by finite element analysis of ANSYS software.

As seen from the figure above, the maximum stress appears at the fixed end of the cantilever beam. The maximum stress of the cantilever beam is 28.5MPa. The maximum stress of the cantilever beam is 27.9MPa when the die is removed. The maximum stress in both conditions is less than the yield limit of the cantilever beam, and the cantilever beam meets the working requirements. Use the above analysis to solve the save as command flow, and then input the random variables, extract the maximum stress, and set the limit state equation.

Because of the large size of the cantilever beam, only a few key structural parameters are considered, and the material and load are considered random variables. The modulus of elasticity, yield strength, and density of the material are considered random variables, and the coefficient of variation is 0.1. The Poisson ratio is located at the constant [21]. The random parameters need to be discretized by the “$3\sigma $” principle, and the input and output random variables are shown in Table 2.

The above variables were input into ANSYS for the Monte Carlo sampling simulation, and the sampling method was the Latin hypercube sampling method, which was simulated 1000 times. The sample curve of g(x) is stable, indicating that the simulation sample is sufficient[22], as shown in Figure 10. The sensitivity of the variable is viewed after entering, and the sensitivity of the input variable is shown in Figure 11.

As seen from Figure 12, the parameters with greater influence on the limit state function are ${{\sigma }_{x}},$ $E, $ ${{H}_{1}},$ and ${{H}_{4}}.$

The sensitivity ratio of each parameter is shown in Table 3.

By sensitivity analysis of input variables, four input variables with high correlation are obtained. In the process of fitting the response surface with the four input variables as output, the remaining seven random variables are the input variables. Using the CCD sampling method to sample iteration, the system will determine the sampling frequency according to the input parameters. After 50 sampling simulations, the response surface of the limit state function is obtained, as shown in Figure 13.

The fitting response surface was simulated by 1000 Monte Carlo sampling. To check whether the sampling is convergent, the histogram of the sample ${{D}_{2}}$ is shown in Figure 14. The sample converges to the mean from the graph, indicating that the data sampling is sampled according to the given distribution, and the results are reliable.

The failure rate of the cantilever beam is obtained by sampling simulation of the fitting response surface, as shown in Figure 15.

As seen from Figure 15, the failure probability of the limit state function of cantilever beam is 0.0170879, while the reliability is about 0.9289.

In this paper, the reliability of the cantilever beam’s structural strength is analyzed. Based on the PDS analysis module of ANSYS, the failure probability of the cantilever beam under actual working conditionsis obtained. When the confidence level is 95%, the structural reliability is 0.9289. The sensitivity analysis of the influence parameters of the cantilever beam shows that the width of the cantilever and the length of the free end have great influence on the reliability of the structure, which provides a direction for the next structural optimization. In the structural reliability analysis, the paper combines the advantages of the Monte Carlo method and response surface method and uses a hybrid stochastic finite element method to analyze the reliability of the structure, so as to provide reference for other similar products’ reliability analysis.

This paper is supported by the Inner Mongolia Science & Technology Plan Project and the Inner Mongolia Autonomous Region Science & Technology Innovation to Guide the Reward Project (No. 102-413128).