Keywords

1 Introduction

The traditional safety factor method is still used in the calculation of the strength, stiffness and stability of the material mechanical rods. This method has obvious disadvantages. First, there is no quantitative consideration of the randomness of load effect, material strength and component size, often based on engineering experience or a measured value, so it is inevitable to have human factors, even subjective assumptions; Secondly, safety factor \(k = {{m_{R} } \mathord{\left/ {\vphantom {{m_{R} } {m_{s} }}} \right. \kern-0pt} {m_{s} }}\) depends only on the relative positions of R and S, and depends on their dispersion degree \(\left( {\sigma_{R} ,\sigma_{S} } \right)\) has nothing to do [1], This is not consistent with objective reality. In recent years, there have been some reliability analyses of member bar under basic deformation, but the correlation of failure modes of strength, stiffness and stability has not been considered comprehensively, which is somewhat biased with the actual situation. In this paper, the reliability calculation mode of the component’s online elastic range strength, stiffness and stability is considered simultaneously and the different distribution of random variables is considered. JC method is used to calculate its single reliability, and then the correlation of failure mode is considered, and the general limit range of member bar failure probability is obtained.

2 Equation of Limit State under Basic Deformation of Online Elastic Range of Member Bar

2.1 Selection of Functional Functions, Constants and Random Variables (Taking Plastic Materials as an Example) [2]

Axial Stretching. Generally, it is controlled by strength conditions. Take circular rod as an example,and its function function is

$$g = \frac{1}{4}\pi d^{2} \sigma_{s} - F$$
(1)

Select \(\frac{1}{4}\pi = A\), and diameter \(d = x_{1}\), yield limit \(\sigma_{s} = x_{2}\), pull \(F = x_{3}\), then

$$g = Ax_{1}^{2} x_{2} - x_{3}$$
(2)

Axial Compression.

Take the round rod for example:

  1. 1)

    Large flexibility bar \(\left( {\lambda \ge \lambda_{p} } \right)\) is generally controlled by stable conditions, and its function is

\(g = \pi^{2} EI_{\min } - \left( {\mu L} \right)^{2} F\) namely \(g = \tfrac{{\pi^{3} E}}{64}d^{4} - \mu^{2} l^{2} F\).

Take \(\frac{{\pi^{3} E}}{64} = A\), \(d = x_{1}\), length coefficient \(\mu^{2} = B\), the length \(l = x_{2}\), \(F = x_{3}\), then

$$g = Ax_{1}^{4} - Bx_{2}^{2} x_{3}$$
(3)

In the large flexibility bar \(\lambda = \frac{{4\mu m_{L} }}{{m_{d} }}\), corresponding to the proportional limit flexibility \(\lambda_{p} = \sqrt {\frac{{\pi^{2} E}}{{\sigma_{P} }}}\), where the length coefficient is \(\mu\), and the mean length is \(m_{L}\), mean diameter \(m_{d}\), elastic modulus E, proportional limit \(\sigma_{p}\) are obtained from statistics, so all are taken as constants.

  1. 2)

    Medium flexibility \(\left( {\lambda_{p} > \lambda > \lambda_{S} } \right)\) is generally controlled by stability conditions, and its functional function is

\(g = \left( {a - b\lambda } \right)\frac{{\pi d^{2} }}{4} - F\) namely \(g = \frac{a\pi }{4}d^{2} - \pi \mu bLd - F\).

Take \(\frac{a\pi }{4} = A\), \(d = x_{1}\), \(- \pi \mu b = B\), \(L = x_{2}\), \(d = x_{1}\), \(F = x_{3}\) then

$$g = Ax_{1}^{2} + Bx_{1} x_{2} - x_{3}$$
(4)

In the middle compliance bar \(\lambda_{y} = \frac{{a - \sigma_{S} }}{b}\), \(\lambda_{S}\) is the compliance corresponding to the yield limit, a and b are the coefficients related to materials, which are obtained from statistics, so they are taken as constants.

  1. 3)

    The small flexibility bar \(\left( {\lambda \le \lambda_{S} } \right)\) is generally controlled by strength conditions and its functional function is

    $$g = \frac{1}{4}\pi d^{2} \sigma_{S} - F$$
    (5)

Take \(\frac{\pi }{4} = A\), \(d = x_{1}\), \(\sigma_{S} = x_{2}\), \(F = x_{3}\), then

$$g = Ax_{1}^{2} x_{2} - x_{3}$$
(6)

Use Rivets and Ordinary Bolts for Tension and Compression Connection.

Generally controlled by shear and extrusion strength conditions, the tensile strength conditions of the motherboard are solved by structural measures, and its functional function is

$$g_{1} = n_{1} n_{2} \frac{1}{4}\pi d^{2} \tau_{S} - F$$

Type: \(n_{1}\) is the number of shear surfaces of a rivet or bolt; \(n_{2}\) is the number of rivets or bolts.

Take \(n_{1} n_{2} \frac{\pi }{4} = A\), \(d = x_{1}\), shear yield strength \(\tau_{S} = x_{2}\), \(F = x_{3}\), then

$$g_{1} = Ax_{1}^{2} x_{2} - x_{3}$$
(7)
$$g_{2} = n_{2} d\sum {t \cdot \sigma_{jy} - F}$$

Take \(n_{2} = A\), \(d = x_{1}\), \(\sum {t = x_{2} }\) is the minimum plate thickness,and the extrusion yield limit \(\sigma_{jy} = x_{3}\), \(F = x_{4}\), then

$$g_{2} = Ax_{1} x_{2} x_{3} - x_{4}$$
(8)

Pure Torsion.

It is generally controlled by strength and stiffness conditions, and its functional function is

$$g_{1} = \frac{{\pi d^{3} }}{16}\tau_{S} - M_{T}$$
(9)

Take \(\frac{\pi }{16} = A,\)\(d = x_{1}\), \(\tau_{s} = x_{2}\), torque \(M_{T} = x_{3}\), then

$$g_{1} = Ax_{1}^{3} x_{2} - x_{3}$$
(10)
$$g_{2} = GI_{p} \cdot \frac{\pi }{{180^{ \circ } }}\left[ \theta \right] - M_{T} l = \frac{{\pi^{2} G\left[ \theta \right]}}{5760}d^{4} - M_{T} l$$

Take \(\frac{{\pi^{2} G\left[ \theta \right]}}{5760} = A\), where the shear elastic modulus is obtained by statistics, and \(\left[ \theta \right]\) is given by design requirements, both of which are constants, \(d = x_{1}\), \(M_{T} = x_{2}\), \(l = x_{3}\), then

$$g_{2} = Ax_{1}^{4} - x_{2} x_{3}$$
(11)

Plane Bending.

A simple beam with rectangular section under uniformly distributed load is taken as an example (the analysis method of non-rectangular beam is similar). It is mainly controlled by normal stress and shear stress intensity conditions, and the deformation and stability problems can be solved by structural measures. Its functional function is

$$g_{1} = \frac{1}{6}bh^{2} \sigma_{S} - \frac{1}{8}pl^{2}$$

Take \(\frac{1}{6} = A\), width \(b = x_{1}\), \(h = x_{2}\), \(\sigma_{S} = x_{3}\), \(\frac{1}{8} = B\), load concentration \(p = x_{4}\), \(l = x_{5}\), then

$$g_{1} = Ax_{1} x_{2}^{2} x_{3} - Bx_{4} x_{5}^{2}$$
(12)
$$g_{2} = \frac{2}{3}bh\tau_{S} - \frac{1}{2}pl$$

Take \(\frac{2}{3} = A\), width \(b = x_{1}\), \(h = x_{2}\), \(\tau_{S} = x_{3}\), \(\frac{1}{2} = B\), load concentration \(p = x_{4}\), \(l = x_{5}\) then

$$g_{2} = Ax_{1} x_{2}^{{}} x_{3} - Bx_{4} x_{5}$$
(13)

2.2 Equation of Limit State

In structural reliability analysis, the limit state of the structure is generally the case that the function function is equal to 0, namely

$$Z = g(x_{1} ,x_{2} ,...,x_{n} ) = 0$$

For the basic deformation of the rod, the limit state is (1) to (13) equal to 0.

3 Reliability Calculation Mode in the Case of Different Distribution of Random Variables and Correlation of Failure Modes

3.1 JC Method's Equivalent Normal Mean and Standard Deviation and Component Failure Probability \(p_{f}\) the Bounds Interval Estimation Method

JC method is suitable for solving structural reliability index under random distribution of random variables. The basic principle of this method is to replace the non-normal distribution random variables with normal distribution, but it requires to replace the normal distribution function in the design of checking point \(x_{i}^{ * }\), the cumulative probability distribution function (CDF) and probability density function (PDF) values are the same as the original distribution function (CDF)and (PDF) values According to the above two conditions, The mean and standard deviation of the equivalent normal distribution (\(\overline{X}^{\prime}_{i}\) and \({\sigma }_{{X}_{i}^{^{\prime}}}\)) are calculated according to the above two conditions. Finally, the reliability index and failure probability of components are calculated by the second-order matrix method. If a component is designed with two or more functional functions and the functional functions have the same random variables, there is a correlation between failure modes. and the following formula can be used to find the limit range of component failure probability [3].

$$\mathop {\max }\limits_{i} p_{{f_{i} }} \le p_{f} \le 1 - \prod\limits_{i = 1}^{n} {\left( {1 - p_{{f_{i} }} } \right)}$$
(14)

The probability density functions corresponding to random variables commonly used in engineering structural component design are normally normal distribution lognormal distribution, extreme value I type distribution, \(\Gamma\) distribution, etc. The mean and standard deviation of their equivalent normal distribution are listed below [4]

  1. 1)

    the variable \(x_{i}\) is normal distribution, directly take the standard deviation and mean of the variable as “equivalent normal” standard deviation and mean, namely

    $$\sigma^{\prime}_{{X_{i} }} = \sigma_{{X_{i} }}$$
    (15)
    $$\overline{X}_{i}^{^{\prime}} = m_{{x_{i} }}$$
    (16)
  1. 2)

    the variable \(x_{i}\) for lognormal distribution, the standard deviation and mean of the equivalent normal distribution are

    $$\sigma^{\prime}_{{X_{i} }} = x_{i}^{ * } \left[ {\ln \left( {1 + v_{{x_{i} }}^{2} } \right)} \right]^{\frac{1}{2}}$$
    (17)
    $$\overline{X}^{\prime}_{i} = \overline{X}_{i}^{ * } - s_{i}^{ * } \sigma^{\prime}_{{X_{i} }}$$
    (18)

Type: \(s_{i}^{ * } = \frac{{\ln x_{i}^{ * } - \left[ {\ln m_{{x_{i} }} - \frac{1}{2}\ln \left( {1 + v_{{x_{i} }} } \right)^{2} } \right]}}{{\left[ {\ln \left( {1 + v_{{x_{i} }}^{ * } } \right)} \right]^{\frac{1}{2}} }}\)

Type: \(m_{{x_{i} }}\) is mean; \(v_{{x_{i} }}\) is the coefficient.

  1. 3)

    the variable \(x_{i}\) is the extreme value I distribution, its equivalent normal distribution standard deviation and mean is h

    $$\sigma_{{X_{i} }}^{^{\prime}} = {{\varphi \left[ {\Phi^{ - 1} \left( {F_{{x_{i} }} } \right)} \right]} \mathord{\left/ {\vphantom {{\phi \left[ {\Phi^{ - 1} \left( {F_{{x_{i} }} } \right)} \right]} {f_{{X_{i} }} \left( {x_{i}^{ * } } \right)}}} \right. \kern-0pt} {f_{{X_{i} }} \left( {x_{i}^{ * } } \right)}}$$
    (19)
    $$\overline{X}^{\prime} = x_{i}^{ * } - \sigma^{\prime}_{{X_{i} }} \Phi^{ - 1} \left[ {F_{{X_{i} }} \left( {x_{i}^{ * } } \right)} \right]$$
    (20)

Type: \(x_{i}\) is check points for design.

$$f_{{X_{i} }} \left( {x_{i}^{ * } } \right) = a\exp \left[ { - a\left( {x_{i}^{ * } - k} \right) - e^{{ - a\left( {x_{i}^{ * } - k} \right)}} } \right]$$
(21)

is variable \(x_{i}\), the original probability density is the value of the design check point.

$$F_{Xi} \left( {x_{i}^{ * } } \right) = \exp \left[ { - \exp \left( { - a\left( {x_{i}^{ * } - k} \right)} \right)} \right]$$
(22)

is variable \(x_{i}\), the original cumulative probability distribution function is the value of the design check point.

$$a = {{1.2825} \mathord{\left/ {\vphantom {{1.2825} {\sigma_{{X_{i} }} }}} \right. \kern-0pt} {\sigma_{{X_{i} }} }}\,\,\,\,k = m_{{x_{i} }} - {{0.5772} \mathord{\left/ {\vphantom {{0.5772} a}} \right. \kern-0pt} a}$$
  1. 4)

    variable \(x_{i}\) is \(\Gamma\) distribution,the standard deviation and mean of the equivalent normal distribution are

    $$\sigma^{\prime}_{{X_{i} }} = {{\phi \left[ {\Phi^{ - 1} \left( {F_{{x_{i} }} } \right)} \right]} \mathord{\left/ {\vphantom {{\varphi \left[ {\Phi^{ - 1} \left( {F_{{x_{i} }} } \right)} \right]} {f_{{X_{i} }} \left( {x_{i}^{ * } } \right)}}} \right. \kern-0pt} {f_{{X_{i} }} \left( {x_{i}^{ * } } \right)}}$$
    (23)
    $$\overline{X}^{\prime}_{i} = x_{i}^{ * } - \sigma^{\prime}_{{X_{i} }} \Phi^{ - 1} \left[ {F_{{x_{i} }} \left( {x_{i}^{ * } } \right)} \right]$$
    (24)

Type: \(f_{{X_{i} }} \left( {x_{i}^{ * } } \right) = \frac{{\lambda \left( {\lambda x_{i}^{ * } } \right)^{k - 1} e^{{ - \lambda x_{i}^{ * } }} }}{\Gamma \left( k \right)}\), \(x_{i} \ge 0\), \(\Gamma \left( k \right) = \int_{0}^{\infty } {e^{ - u} u^{k - 1} } du\); \(\lambda\) and k are two parameters that can be derived from the random variable \(x_{i}\) the original mean \(\mu\) standard deviation \(\sigma\) formula find out. Formula: \(\mu = {k \mathord{\left/ {\vphantom {k \lambda }} \right. \kern-0pt} \lambda }\),

$$F_{X} \left( {x_{i}^{ * } } \right) = \frac{1}{\Gamma \left( k \right)}\int_{0}^{{\lambda x_{i}^{ * } }} {e^{ - u} u^{k - 1} } du$$
(25)

\(f_{X} \left( {x_{i}^{ * } } \right)\), \(F_{X} \left( {x_{i}^{ * } } \right)\) are the value of the original probability density and accumulated probability distribution function at the design checking point \(x_{i}^{ * }\).

3.2 Jc Method Analysis Steps and Component Failure Probability \(p_{f}\) Bound Interval Estimation Method [5]

Functional functions (1) to (13) JC method analysis steps are the same, axial tension and axial compression rod design only one failure mode; And rivets, ordinary bolt connection, pure torsion, plane bending have two related failure modes. The latter analysis steps include the former, and the latter failure probability JC method analysis steps, failure probability \(p_{f}\) the estimation methods of the boundary interval are the same [6], so the rivets or ordinary bolts are used as examples to illustrate.

1) JC method is used to analyze the limit state equation of function function (7) and obtain its reliable index \(\beta\):

① Assume a beta value \(\beta\).

②For all \(i\) values, the initial value of the design checking point is selected \(x_{i}^{ * } = m_{{x_{i} }}\).

③Calculate \(\sigma^{\prime}_{{x_{i} }}\) and \(\overline{X}^{\prime}_{i}\) (The formulas for \(\sigma^{\prime}_{{x_{i} }}\) and \(\overline{X}^{\prime}_{i}\) for various basic deformation have been found)by \(\sigma^{\prime}_{{x_{i} }} = \varphi {{\left[ {\Phi^{ - 1} \left( {F_{{x_{i} }} } \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {\Phi^{ - 1} \left( {F_{{x_{i} }} } \right)} \right]} {f_{{X_{i} }} }}} \right. \kern-0pt} {f_{{X_{i} }} }}\left( {x_{i}^{ * } } \right)\),\(\overline{X}^{\prime}_{i} = x_{i}^{ * } - \sigma^{\prime}_{{X_{i} }} \Phi^{ - 1} \left[ {F_{{x_{i} }} \left( {x_{i}^{ * } } \right)} \right]\).

④Calculate \({{\partial_{g} } \mathord{\left/ {\vphantom {{\partial_{g} } {\partial_{{x_{i} }} \left| {_{{x^{ * } }} } \right.}}} \right. \kern-0pt} {\partial_{{x_{i} }} \left| {_{{x^{ * } }} } \right.}}\), taking Eq. (7) as an example

$$\frac{{\partial_{{g_{1} }} }}{{\partial x_{1} }} = 2Ax_{1} x_{2} \,\,\,\frac{{\partial_{{g_{1} }} }}{{\partial x_{2} }} = Ax_{1}^{2} = - 1$$

⑤Calculate the sensitivity coefficient \(a_{i}\) from \(a_{i} = \frac{{\sigma^{\prime}_{{x_{i} }} \cdot \frac{{\partial_{g} }}{{\partial x_{i} }}\left| {_{{x_{i}^{ * } }} } \right.}}{{\sqrt {\sum\limits_{i = 1}^{n} {\left( {\sigma^{\prime}_{{x_{i} }} \cdot \frac{{\partial_{{g_{1} }} }}{{\partial x_{i} }}\left| {_{{x_{i}^{ * } }} } \right.} \right)^{2} } } }}\) \(\left( {i = 1,2,3} \right)\).

⑥Calculate the new design checking point \(X_{I}^{ * } = \overline{X}^{\prime} - a_{i} \beta \sigma^{\prime}_{{x_{i} }}\). Repeat steps ③ ~ ⑥ until \(x_{i}^{ * }\) the two differences are within the allowable range.

⑦Take \(x_{i}^{ * }\) into the limit state equation \(g\left( {x_{i}^{ * } } \right) = 0\), calculate the value of \(\beta\), take the limit state equation of function (7) as an example:

$$g_{1} = C\beta^{2} - D\beta + E = 0$$

Type: \(C = Aa_{1} a_{2} \sigma^{\prime}_{{x_{1} }} \sigma^{\prime}_{{x_{2} }} ;\,D = Aa_{1} \sigma^{\prime}_{{x_{1} }} \overline{X}^{\prime}_{2} + Aa_{2} \sigma^{\prime}_{{x_{2} }} \overline{X}^{\prime}_{2} - a_{3} \sigma^{\prime}_{{x_{3} }} ;\,\,E = A\overline{X}^{\prime}_{1} \overline{X}^{\prime}_{2} - \overline{X}^{\prime}_{3}\), then \(\beta = \frac{{D \pm \sqrt {D^{2} - 4CE} }}{2C}\),take a reasonable value of \(\beta\).

⑧Repeat steps ③ ~ ⑦ until the absolute value of \(\beta\) difference is very small \(\left( {\Delta \beta \ge 0.01} \right)\). Finding the failure probability \(p_{f}\) by \(p_{f} = 1 - \Phi \left( \beta \right)\).

2) The general limit range of component failure probability when the limit state function is related.

$$\mathop {\max }\limits_{i} p_{{f_{i} }} \le p_{f} \le 1 - \prod\limits_{i = 1}^{n} {\left( {1 - p_{{f_{i} }} } \right)}$$

The limit state equations corresponding to functional functions (7) and (8), (10) and (11), (12) and (13) are related by random variables respectively, and the reliability limit under basic deformation of components can be calculated from the above equation [7].

4 Conclusion

In this paper, the failure modes of strength stability and stiffness under basic deformation in the online elastic range of member bar are comprehensively analyzed. the equivalent normal mean and standard deviation of JC method are calculated when random variables are distributed differently, and the reliability limit analysis method of member bar is given when functional functions are related. The results provide a preliminary conclusion for further analysis of reliability calculation of member bar under basic deformation in the elastic–plastic range,and further analysis of reliability calculation of member bar under combined deformation in the online elastic range and elastic–plastic range. It can be used for reference in the analysis of structural reliability.