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A new probabilistic finite element calculation method

classification number: tb115 document identification code: a

article number: (2000) Abstract: according to the basis theory of probability finite element method, on the conventional second moment analysis technique, this paper put forward the second moment. They all expressed their views, opinions and suggestions on the ways to solve the problem The feasibility of the new method has been proved through the Monte-Garlo method imitating the consequence. It is certified that that the second is very strict with all indicators of the experimental machine d-moment differential method is a new feasible method

Key Words：probability finite element method; the second-moment analysis technique; The second moment differential method ▲ finite element method is a numerical analysis method for solving approximate solutions of many engineering problems. Since there are many ready-made program systems, they have been widely used in almost all engineering fields. However, the finite element method currently used is a traditional calculation method. With the progress of science and technology and the development of reliability, the traditional finite element method has been expanded further to form a finite element method considering that the design variables are random variables, that is, probabilistic finite element method. Using this method, the structural reliability of complex systems can be calculated more accurately, and the distribution of other mechanical properties can be obtained at the same time

some references [1] to [3] have proposed the finite element method and used it to solve the problems of simple structures. However, the current probabilistic finite element method is only theoretically analyzed because it takes up a lot of time and memory. Now there are also some simplified probabilistic finite element methods combined with the first-order second moment theory [4]. On this basis, a new algorithm, first-order second-order moment differential method, is proposed. Practice has proved that it is a better approximate calculation method. The first-order second-order moment differential method first-order second-order moment differential method is derived from the basic theory of probability finite element method and Taylor formula as the guide to simplify the correlation matrix solution method and implicit derivative method in probability finite element method

assuming that R represents the structural strength and s represents the stress in the structure, and both are random variables, the structural reliability is r=p{r-s> 0} (1) if G (x) = R-S, G (x) is the critical function of the structure, where x=xi (i=1,2... M) is many basic random variables affecting R and S. at this time, formula (1) can be expressed as r=p{g (x)> 0} (2) in order to calculate R, the distribution function of G (x) must be known first, and the distribution of G (x) is difficult to be determined according to the distribution of Xi. Here, assuming that G (x) and Xi obey the normal distribution, and the mean and variance are e (XI), e (g), V (XI), V (g), then there is [g obeys the standard normal distribution, and (3) order, then there is r=1- φ (- β)=φ ( β) (4) Among them φ (x) - is the standard normal cumulative distribution function

β— Is the reliability coefficient

if required β， E (g) and V (g) should be calculated first. Because the critical function g (x) is usually nonlinear, it is difficult to directly calculate the mean and variance of G (x) from the mean and variance of Xi. G (x) can be expanded according to Taylor formula at the design verification point x* (5) because the coefficient of variation of each random variable is small, more than the second derivative term can be omitted, so (6) (7) for example, the critical function of bending fatigue strength of gear root can be expressed as G (x)= σ L- σ Eff (8), where:

σ L - fatigue strength limit

σ Eff - equivalent stress

we choose power P, speed n, gap Δ、 Fatigue strength limit σ L and Yang's can't form the cooperation modulus E as the basic random variable when competing with the outside world. At this time, (9) (10) is obtained from equation (8), and both sides of the equation apply this random variable to a wide range of customers because of its wide range of use σ B and P, N Δ、 There is an implicit function relationship between E and other variables

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