Reliability sensitivity numerical analysis of mechanical structure based on gamma processes

Abstract

Using random variables to describe uncertain parameters in structural systems, its initial strength and the evolution process of the strength degradation is regarded as the Gamma process. In this article, we propose a new method on reliability sensitivity numerical analysis of mechanical structure based on Gamma processes. Then, we use the fourth moment method based on frequency curve of Pearson to solve the problem of reliability calculation with random parameters of arbitrary distributions. Formulas for calculating the reliability sensitivity with respect to the mean and the variance of the random variables are derived. The reliability analysis of the welded box girders of crane is taken as an example to verify the proposed method. The results show that the method can effectively solve the problem of the reliability sensitivity of structural systems with strength degradation.

Keywords

Sensitivity analyses strength of welded structures strength degradation moment method frequency curve of Pearson

Introduction

The traditional mechanical structural reliability models often do not consider the structural strength degradation, thus resulting obtained reliability is a constant value.^1–3 In practical engineering, the long service of the mechanical structure leads to wear and tear of the material, time-varying of load effects, gradual decrease in structural strength, and a variety of other factors lead to gradient properties of structural reliability and failure probability. Therefore, the real situation can be reflected more actually by the gradient reliability, especially given that the gradient reliability of the structural strength degradation is very important for objective assessment of the reliability of mechanical structures and systems and to the safe operation, rational development, and maintenance projects of the entire system.⁴ Andrieu-Renauda and Lemairea⁵ put forward an efficient method to solve time-varying reliability on the basis of the analysis techniques of system reliability; the effects of degradation are considered by a stress intensity interference model given by Lewis and Chen;⁶ however, it only analyzes the situation that considers deterministic stress and stochastic intensity degradation or random stress and deterministic intensity degradation; when considering the independence between the stress and the strength degradation, Deng and Gu⁷ studied stress intensity reliability problems under the discrete stochastic stress and strength degradation. In fact, the strength degradation of the product is usually caused by the effect of stress; therefore, they are not independent. Chen and Shen⁸ established a numerical analysis method which can simulate the steel structure and structural strength degradation thereof. Qin and Yang⁹ established the calculation model for time-varying reliability of the existing bridge and used adaptive importance sampling method to calculate the time-varying reliability index. The above research is carried out in the premise of obtaining the exact distribution of the structural parameters. For this reason, this article tries to describe the mechanical structural strength degradation with Gamma random process; on this basis, the reliability index and reliability sensitivity are deduced based on the fourth-order moment method of Pearson frequency curves. The sensitivity variation of random variables was studied on the basis of reliability analysis, and the curve of reliability and sensitivity was mapped by taking the metal structure of the $DQ 75 t - 28 m$ bridge crane as an example.

Structural strength degradation model of Gamma process

Mechanical structural strength degradation is in one way diminishing. Gamma degradation process is a random process subjected to the same size parameter Gamma distribution and having independent non-negative increments, which is suitable for modeling the damage that accumulates over time a series of small increments, such as structural fatigue, wear, crack growth, and creep.^10,11 Random degradation process of the structure strength is considered to be a Gamma process that can ensure the monotone increasing of the degradation.

Definition of Gamma random process

Suppose the probability density function of the random variable X is

G (x | v, μ) = {\begin{matrix} \frac{μ^{v}}{Γ (v)} x^{v - 1} e^{- μ x}, & x \in [0, + \infty] \\ 0, & else \end{matrix}

(1)

where $Γ (v) = \int_{z = 0}^{\infty} z^{v - 1} e^{- z} dz, v > 0$ is Gamma function, v is a shape parameter (the value is greater than 0), and $μ$ is a size parameter (the value is less than 0). And the random variable X meets the Gamma distribution.^12–16

When $t \geq 0$ , shape parameter $v (t)$ is a non-decreasing right continuous real function, and $v (0) \equiv 0$ . Gamma distribution is a random process with continuous time ${X (t), t \geq 0}$ , and there are three mathematical properties as follows:

$X (0) = 0$ ;

$\forall τ \geq 0, X (τ) - X (t) ~ G_{a} (v (τ) - v (t), μ)$ ;

$X (t)$ has independent increments, $0 < t_{0} < t_{1} < \dots < t_{n} < t_{n + 1}$ , $X (t_{1}) - X (t_{0}), X (t_{2}) - X (t_{1}), …, X (t_{n + 1}) - X (t_{n})$ are independent.

Mean and variance of Gamma random process

The degradation amount of structural strength at $t (t \geq 0)$ is described with $X (t)$ , the degradation amount of structural strength at t, $X (t)$ , meets the Gamma distribution, where in its scale parameter $μ$ does not change with time, the shape parameter $v (t)$ is a time-varying parameter, then it can be seen according to formula (1) that the probability density function of $X (t)$ is

f_{X (t)} (x) = G (x | v (t), μ)

(2)

The mean value and variance of degradation amount of structural strength at t is

{\begin{cases} E [X (t)] = \frac{v (t)}{μ} \\ V a r [X (t)] = \frac{v (t)}{μ^{2}} \end{cases}

(3)

We consider degradation as a stationary gamma process with $v (t) = c t^{b}$ . Experimental studies show¹⁷ that the expectation of degradation at t is often proportional to power law of the time, namely

E [X (t)] = \frac{v (t)}{μ} = \frac{c t^{b}}{μ} = a t^{b} \propto t^{b}, a, b, c > 0, b \neq 1

(4)

Parameter estimation of Gamma random process

Gamma function $F (v)$ with shape function $v (t) = c t^{b}$ and dimension parameter, we suppose that $μ$ is known, c and $μ$ is unknown. Some examples of expected deterioration according to a power law are the expected degradation of concrete due to corrosion of reinforcement (linear: $b = 1$ ),¹⁸ sulfate attack (parabolic: $b = 2$ ),¹⁸ diffusion-controlled aging (square root: $b = 0.5$ ),¹⁸ creep ( $b = 1 / 8$ ),¹⁹ and the expected scour-hole depth $(b = 0.4)$ .^20,21 The gamma process is called stationary if the expected deterioration is linear in time, that is, when $b = 1$ in equation (4), and non-stationary when $b \neq 1$ . Their values are obtained from logarithmic maximum likelihood estimator of the likelihood function of statistics;¹⁹ the specific process is as follows.

Suppose there are m samples used for structural strength degradation test, they are tested at $t_{i} (i = 1, 2, \dots, n, 0 = t_{0} < t_{1} < t_{2} < \dots < t_{n})$ . The value of the cumulative degradation amount of the $i (th)$ measurement of the $j (th) (j = 1, 2, \dots, m)$ sample can be obtained via a degradation test. The mean and variance of the product degradation amount of the $i (th)$ measurement can be obtained by formula (3). Considering that the degradation increment $θ_{i} = x_{i} - x_{i - 1} (i = 1, 2, \dots, n)$ meets the Gamma distribution, it can be obtained that the log-likelihood function is

L (θ_{1}, θ_{2}, …, θ_{n} | c, μ) = \prod_{i = 1}^{n} f_{X (t_{i}) - X (t_{i - 1})} (θ_{i}) = \prod_{i = 1}^{n} \frac{μ^{c [t_{i}^{b} - t_{i - 1}^{b}]}}{Γ (c [t_{i}^{b} - t_{i - 1}^{b}])} θ_{i}^{c [t_{i}^{b} - t_{i - 1}^{b}] - 1} e^{- μ θ_{i}}

(5)

{\begin{array}{l} \hat{μ} = \frac{\hat{c} t_{n}^{b}}{x_{n}} \\ \sum_{i = 1}^{n} [t_{i}^{b} - t_{i - 1}^{b}] {ϕ (\hat{c} [t_{i}^{b} - t_{i - 1}^{b}]) - \log θ_{i}} = t_{n}^{b} \log (\frac{\hat{c} t_{n}^{b}}{x_{n}}) \end{array}

(6)

As for the first partial derivative of the log-likelihood function, the maximum likelihood estimator $\hat{c}$ and $\hat{μ}$ can be obtained by calculating the first-order derivative of a likelihood equation, thus the expectation of the structural strength degradation can be expressed as follows

E (X (t)) = x_{n} {[\frac{t}{t_{n}}]}^{b}

(7)

Structure performance function of the failure probability with fourth moment calculation method

Structural reliability design perturbation method

In the process of mechanical structure reliability design, structure function (limit state functions) as a function of the basic variables by $x_{1}, x_{2}, \dots, x_{n}$ can be expressed as

Z = R - S = g (X), X = (x_{1}, x_{2}, …, x_{n})

(8)

where R is the structural effect and S is the load effect. They are main target of the reliability study of the mechanical structure. Functions between the two determine the reliability of the mechanical structure. The basic random variable $X = (x_{1}, x_{2}, \dots, x_{n})$ is to characterize the design size, material properties, loads, and so on. n is a dimension affecting structural reliability parameters. According to the random perturbation theory, stochastic parameter vector X and state function $g (X)$ can be expanded as

{\begin{cases} X = X_{d} + Δ X_{p} \\ g (X) = g_{d} (X) + Δ g_{p} (X) \end{cases}

(9)

where the subscript d represents the determined part of the random variable, and the subscript p represents the random part of the random variable. Both of them have zero mean, and the value of the random part is much smaller than that of the determined part. $Δ$ is infinitesimal of high order. According to stochastic analysis theory, the mean of formula (9) is

{\begin{matrix} E (X) = \bar{X} = E (X_{d}) + Δ E (X_{p}) = X_{d} \\ μ_{g} = E [g (X)] = E [g_{d} (X)] + Δ E [g_{p} (X)] = g_{d} (X) \end{matrix}

(10)

Applying algebra theory of Kronecker and stochastic analysis theory, the following three expressions can be obtained by taking the second, third, and fourth moment of formula (10) on both sides

\begin{matrix} α_{2 g} = σ_{g}^{2} = Var [g (X)] = Δ^{2} E [{(\frac{\partial g_{d} (X)}{\partial X^{T}})}^{[2]} X_{p}^{2}] \\ = {(\frac{\partial g_{d} (X)}{\partial X^{T}})}^{[2]} Var (X) \end{matrix}

(11)

α_{3 g} = Δ^{3} E [{(\frac{\partial g_{d} (X)}{\partial X^{T}})}^{[3]} X_{p}^{3}] = {(\frac{\partial g_{d} (X)}{\partial X^{T}})}^{[3]} C_{3} (X)

(12)

α_{4 g} = Δ^{4} E [{(\frac{\partial g_{d} (X)}{\partial X^{T}})}^{[4]} X_{p}^{4}] = {(\frac{\partial g_{d} (X)}{\partial X^{T}})}^{[4]} C_{4} (X)

(13)

where the power of Kronecker. $Var (X)$ , $C_{3} (X)$ and $C_{4} (X)$ are second moment, third moment, and fourth moment vector of the random variable, respectively. $α_{2 g}, α_{3 g}, and α_{4 g}$ are second moment, third moment, and fourth moment of the structural performance function $Z = g (X)$ .

The partial derivative of structural performance function $Z = g (X)$ for the basic random variable X is

\frac{\partial g (X)}{\partial X^{T}} = {[\frac{\partial g (X)}{\partial X_{1}} \frac{\partial g (X)}{\partial X_{2}} \dots \frac{\partial g (X)}{\partial X_{n}}]}_{1 \times n}

(14)

The second moment, third moment, and fourth moment can be obtained by substituting formula (14) into formulas (11), (12), and (13).

Based on Pearson frequency curves’ fourth moment method

When the skewness $α_{3 g}$ of the performance function $Z = g (X)$ is relatively large, computed results of the fourth moment method based on high-order moment standardization technique (HOMST) often have large error.²² The probability density function obtained on the basis of fourth moment method of Edgeworth series sometimes cannot meet the definition of the probability density function; the obtained reliability may be larger than one, which obviously does not meet the requirements of reliability analysis.²³ This article selects the fourth moment method based on Pearson frequency curves; this method selects the most appropriate curve from a cluster of curves as the probability density function of the structural performance function.

Formulas (11)–(13) are standardized by $z_{u} = (z - μ_{g}) / σ_{g}$ . In the Pearson system, the probability density function of the standardized variable satisfies the differential equations as follows^24–27

\frac{1}{f} \frac{df}{d z_{u}} = - \frac{a z_{u} + b}{c + b z_{u} + {dz}_{u}^{2}}

(15)

where $a, b, c$ and d can be expressed by the first four central moments, that is

\begin{matrix} a = 10 α_{4 g} - 12 α_{3 g}^{2} - 18 \\ b = α_{3 g} (α_{4 g} + 3) \\ c = 4 α_{4 g} - 3 α_{3 g}^{2} \\ d = 2 α_{4 g} - 3 α_{3 g}^{2} - 6 \end{matrix}

(16)

Because

P {z \leq 0} = P {z \leq - \frac{μ_{g}}{σ_{g}}} = P {z_{u} \leq - β_{2 M}}

(17)

The corresponding fourth moment reliability index derived from formulas (15)–(17) is

β_{4 M} = - Φ^{- 1} (\int_{- \infty}^{- β_{2 M}} f (z_{u}) d z_{u})

(18)

Mechanical structure deduced reliability sensitivity

According to formula (18), the sensitivity calculation formula of $P_{f}$ (failure probability) to X (basic variable) when considering the fourth moments can be derived from the relationship between $P_{f}$ and $β$ (reliability index), as well as the relationship between $β$ and the moments of $Z = g (X)$ (limit state function) using a composite function derivation method

\frac{\partial P_{f}}{\partial {\bar{X}}^{T}} = \frac{\partial P_{f}}{\partial β_{4 M}} \frac{\partial β_{4 M}}{\partial {\bar{X}}_{T}} = \frac{\partial P_{f}}{\partial β_{4 M}} [\frac{\partial β_{4 M}}{\partial β_{2 M}} (\frac{\partial β_{2 M}}{\partial μ_{g}} \frac{\partial μ_{g}}{\partial {\bar{X}}^{T}} + \frac{\partial β_{2 M}}{\partial σ_{2 g}} \frac{\partial α_{2 g}}{\partial {\bar{X}}^{T}}) + \frac{\partial β_{4 M}}{\partial σ_{3 g}} \frac{\partial σ_{3 g}}{\partial {\bar{X}}^{T}} + \frac{\partial β_{4 M}}{\partial σ_{4 g}} \frac{\partial σ_{4 g}}{\partial {\bar{X}}^{T}}]

(19)

where

\frac{\partial β_{2 M}}{\partial μ_{g}} = \frac{1}{σ_{g}^{2}} = \frac{1}{σ_{2 g}}

(20)

\frac{\partial β_{2 M}}{\partial α_{2 g}} = - \frac{μ_{g}}{σ_{g}^{2}} = - \frac{α_{1 g}}{α_{2 g}^{2}}

(21)

\frac{\partial P_{f}}{\partial β_{4 M}} = - \frac{1}{\sqrt{2 π}} e^{- \frac{β_{4 M}^{2}}{2}}

(22)

\frac{\partial β_{4 M}}{\partial β_{2 M}} = \frac{3 (α_{4 g} - 1) + 2 α_{3 g} β_{2 M}}{sqrt (5 α_{3 g}^{2} - 9 α_{4 g} + 9) (1 - α_{4 g})}

(23)

\frac{\partial β_{4 M}}{\partial α_{3 g}} = \frac{β_{2 M}^{2} - 1}{sqrt (5 α_{3 g}^{2} - 9 α_{4 g} + 9) (1 - α_{4 g})} + \frac{5 [3 (1 - α_{4 g}) β_{2 M} + α_{3 g} (1 - β_{2 M}^{2})]}{2 \times \sqrt[3]{{[(5 α_{3 g}^{2} - 9 α_{4 g} + 9) (1 - α_{4 g})]}^{2}}} \cdot (1 - α_{4 g}) α_{3 g}

(24)

\frac{\partial β_{4 M}}{\partial α_{4 g}} = \frac{3 β_{2 M} - 1}{s q r t (5 α_{3 g}^{2} - 9 α_{4 g} + 9) (1 - α_{4 g})} - \frac{[3 (1 - α_{4 g}) β_{2 M} + α_{3 g} (1 - β_{2 M}^{2})]}{2 \times \sqrt[3]{{[(5 α_{3 g}^{2} - 9 α_{4 g} + 9) (1 - α_{4 g})]}^{2}}} \cdot (5 α_{3 g}^{2} - 18 α_{4 g} + 18)

(25)

Verification and analysis of cases considered

A metal structure of a DQ $28.5 m_75 t$ welded box double beam bridge crane, as shown in Figure 1, is selected to verify the validity of the proposed method. The main technical parameters and values are shown in Table 1.

Table 1.

Main technical parameters of bridge crane.

Technical parameter	Data	Instruction
Classification group	A6
Rated lifting weight (t)	70	Main vice
Rated lifting weight (t)	15	Main vice
Trolley wheel distance (m)	4.02
Cart wheel distance (m)	4.50
Span (m)	28.5
Rated lifting speed (m/min)	30	Main vice
Rated lifting speed (m/min)	9	Main vice
Trolley speed (m/min)	20
Cart speed (m/min)	50

The main structure of the bridge crane uses Q 235 steel, and the following values of the bridge crane are calculated: the allowable stress $[σ] = 175.37 MPa$ , the elastic modulus $E = 2.06 \times 10^{5} MPa$ , and the material density $ρ = 7850 kg / m^{3}$ .

The following six random variables are confirmed: W (lifting load), B (main beam width), H (main beam height), E (Elastic modulus), $e_YYB$ (flange plate thickness), and $e_FB$ (web thickness), which are listed in Table 2.

Table 2.

Structure random variable of the bridge crane and statical properties.

Random variable	Meaning	Mean	Variance
$W (10^{6} N)$	Lifting load	250	20
H (mm)	Main beam height	8155	366
B (mm)	Main beam width	4108	207
E (MPa)	Elastic modulus	206	10
$e_YYB (mm)$	Flange plate thickness	14	4.2
$e_FB (mm)$	Web thickness	12	2.5

The parameters of the Gamma process are calculated from previous section, $b = 0.6, c = 0.34, μ = 7$ ,^23–26 and the reliability sensitivity thereof is analyzed.

Choosing the case where a fully loaded trolley is located in the middle of bridge crane and the cart is static as the analysis object, the structural failure of the bridge crane occurs when $P_{\max}$ (the maximum stress) is greater than $[σ]$ (the ultimate strength). Suppose $σ$ is the allowable stress of the danger point of bridge crane metal structures, then the limit state function of the bridge crane metal structure can be written as

Z = g (X) = σ - S (W, H, B, E, e_YYB, e_FB)

(26)

Using ISIGHT-FD software to integrate the finite element analysis process, sample data of the random variable stress response can be obtained by the experiment design method.

ISIGHT-FD is used to integrate ANSYS, the above six uncertain parameters are selected, and the Latin hypercube sampling technique is called in the ISIGHT-FD to sample the parameters. Because the six uncertain parameters are selected, only 200 sets of samples need to be input. The output samples of the stress response of the dangerous parts can be obtained by the experimental design, thereby obtaining 200 samples $(X, g (X))$ .

The computing result of the reliability is shown in Figure 2.²⁸ The metal structure reliability of bridge crane considering strength degradation with Monte-Carlo simulation is compared in Figure 2. It can be obtained from Figure 2 that the computing results of both methods of fourth-order moment method are very close. It is acceptable to use this method to calculate the results of the reliability of strength degradation, and the varying pattern of the reliability conforms to the actual engineering situations.

Figures 3 and 4 are the reliability sensitivity of the means and variances of the load W of the bridge crane metal structure, the height H and width B of the main beam, and the elastic modulus E. The variability of these six variables affects the structural reliability rank from large to small: W (lifting load) $> H$ (height of the main beam) $> B$ (width of the main beam) $> E$ (elastic modulus) $> e_YYB$ (flange plate thickness) $> e_FB$ (web thickness). It can be seen from the sensitivity matrix $d R_{i} / dVar (X)$ that the bridge crane metal structure tends to be more unreliable as the basic random parameter covariance increases. However, the changes of the basic random parameter covariance affect the reliability of the mental structure differently.

Figure 5 show the reliability sensitivity with respect to process parameters $b, c, u$ . It can be seen from Figures 3 –5 that the structural reliability of the bridge crane metal structure strength degradation based on Gamma degradation process is most sensitive to the mean and variance of the lifting load, followed by the mean and variance of the main beam height, the least sensitive parameter is the standard deviation and mean of the web thickness. The absolute value of the sensitivity increases with time.

Conclusion

This article combines theoretical analysis with numerical simulation to verify the validity of the fourth moment method based on Pearson frequency curve in gradual reliability of structure on the basis of Monte-Carlo simulation and finite element method. Conclusions can be obtained as follows:

Assuming the structural strength degradation obeys the Gamma random process, a reliability and reliability sensitivity analysis method which considers the strength degradation of mechanical parts is proposed.

In the reliability design of the bridge crane structure, the main ways to improve the reliability of metal structure are to increase the mean of the height of a main beam and reduce the variance of the thickness of a flange plate, and it can also be achieved by reducing the average thickness of the web.

The random parameters b and c of Gamma random process have a great influence on the reliability of parts. The reliability improves with parameter b and reduces with parameter c.

Footnotes

Academic Editor: Yongming Liu

Author’s note

Authors Zhengmao Yang and Yanshan Zhang are equal contributors for the first author in this manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported, in part, by the National Natural Science Foundation of China under Grant No. 61571042 and by the Beijing Higher Education Young Elite Teacher Project (YETP1221).

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