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Based on the Flügge curved beam theory and total inextensible assumption, the dynamic equations of curved pipe’s in-plane vibration are established using the Newton method. The wave propagation method is proposed for calculating the natural frequency of curved pipes with clamped-clamped supported at both ends. Then, the performance function of the resonance reliability of curved pipe conveying fluid is established. Main and total effect indices of global sensitivity analysis (GSA) are introduced. The truncated importance sampling (TIS) method is used for calculating these indices. In the example, the natural frequency and critical velocity of a semicircular pipe are calculated. The importance ranking of input variables is obtained at different working conditions. The method proposed in this paper is valuable and leads to reliability estimation and antiresonance design of curved pipe conveying fluid.

Curved pipes conveying fluid are widely used in aerospace, aircraft, nuclear, petroleum, and ocean engineering. All kinds of uncertainties in fluid velocity, pressure, pipe geometry, and materials will lead to a large change on the natural frequency of the curved pipe conveying fluid through the fluid-structure interaction. When the natural frequency is close to the excitation frequency, this will lead to the resonance failure of pipes. The resonance will have a large damage to the curved pipe. So it is essential and important to study the resonance reliability analysis of curved pipe conveying fluid.

Up to now, there have been a lot of researches in solving the natural frequency of curved pipes conveying fluid [

Although researchers have made a lot of progress in vibration analysis of pipes conveying fluid, the study of resonance failure of pipes conveying fluid is scarce [

The resonance failure is getting more and more serious in engineering especially in the pipe systems. But there are few papers studying the resonance failure of the pipes conveying fluid especially for the curved pipe. Because the curved pipes are widely used in engineering, studying the resonance failure of curved pipes conveying fluid is essential and important. There are many input variables affecting the failure probabilities and the uncertainty of each variable has different effects on the resonance failure probability. The sensitivity analysis aims to measure the uncertainty of each input variable on the failure probabilities including local sensitivities analysis (LSA) and global sensitivity analysis (GSA). Compared to the LSA, the GSA reflects the effect of the variable’s distribution on the failure probabilities globally. Many global sensitivity analyses such as the variance-based GSA and moment independent GSA are available. This work is proposed for reducing the failure probability of curved pipes conveying fluid, so the global reliability sensitivity indices defined by Cui L.J. [

This paper is organized as follows: Section

There are two assumptions of approximating curved pipe, i.e., the total extensible assumption and total inextensible assumption. In this paper, on the base of total inextensible assumption, the Flügge curved beam was applied, and the wave propagation method is proposed for calculating the natural frequency of in-plane vibration.

The force diagram of the pipe is sketched in Figure

The force sketch of curved pipe.

The equilibrium equation along the tangent line is as follows:

The equilibrium equation along the normal line is

The moment equilibrium equation is

Equation (

The force diagram of fluid is sketched as Figure

The force sketch of the fluid.

The relationship of coordinate systems before and after deformation is

Combining

Then, the equilibrium equation along the tangent line can be written as

Equation (

The normal equilibrium equation is

Combining (

In this paper, the Flügge beam model is used to approximate the curved pipe. The axial force

Because the axial strain

The fluid acceleration in the pipe can be described as

Combining (

Tangent dynamic equation:

Normal dynamic equation:

When the motion is simple harmonic vibration, tangential displacement and radial displacement can be written as [

Substituting (

To solve the equation above, the determinant on the left side of (

Equation (

As shown in Figure

The wave mode in semicircular pipe.

where

The transformation matrices of left and right wave propagation are

The reflection relation on the left side is

Combined with (

According to the total inextensible assumption

Substituting (

Substituting (

And the reflect matrix on the right side of the pipe is

The wave propagation and reflection in the pipe can be expressed as follows [

The equation above can be written as

It can be seen that (

In order to avoid resonance behavior, the excitation frequency must be far away from the natural frequency of the curved pipe conveying fluid [

In the reality engineering, due to the uncertainties among materials, manufacturing, installing, and servicing, the excited frequency, natural frequency, and vibration response are random variables. According to the traditional design criterion, there exist risks in resonance design if

The resonance reliability can also be expressed as follows:

Equation (

Suppose the limit state function is given as

The failure domain of this structure system is defined as

Suppose the indicator function of this failure domain is given as

Then the failure probability can be expressed as

A modified version of global reliability sensitivity index defined by Cui in [

Further, Li proved that

In GSA, the global sensitivity indices of a group of input variables are defined as main and total effect indices.

The main effect index of the single input variable is defined as

The total effect index of the single input variables is defined as

It is easy to know that the larger the main effect index

Due to the well-known low total expectation,

In this paper, the TIS procedure is used to calculate the resonance failure probability. Compared with the other procedure, the TIS procedure can further improve the efficiency of solving.

The basic idea of TIS is as follows: in the standard normal space, the most probable point (MPP)

The PDF is truncated by the

We can rewrite (

Furthermore, by inducing the indicator function

Generating N samples according to importance sampling PDF, the mean value of samples can be regarded as the expectation value.

The TIS method just needs to calculate the samples which are in the outer space of the hypersphere. And the traditional method needs to calculate the limit state function values of all sampling points. So the TIS method is more efficient than the traditional method.

Above all, the following five steps based on TIS can be implemented for computing the

Generate

Generate another

Compute the values of functions

Compute

Compute the probable errors of

The curved pipe is sketched in Figure

The distribution parameter of the input variables of example.

Input variables/unit | Distribution | Mean value | variance |
---|---|---|---|

Pipe Yang’s modulus(GPa) | Gauss | 68.9 | 3.43 |

Pipe outer diameter(m) | Gauss | 0.0232 | 0.00116 |

Pipe thickness(m) | Gauss | 0.0041 | 0.000205 |

Pipe density(kg/m3) | Gauss | 7197 | 359.85 |

Fluid density(kg/m3) | Gauss | 1000 | 50 |

Velocity(m/s) | Gauss | 0 | 0.05 |

Maximum excitation frequency(rad/s) | Gauss | 410 | 20.5 |

Minimum excitation frequency(rad/s) | Gauss | 210 | 10.5 |

Radius of curvature(m) | Gauss | 0.5 | 0.025 |

Pressure(MPa) | Gauss | 10 | 0.5 |

The curved pipe illustration of supported.

The first four natural frequencies are listed in Table

The relationship of four natural frequencies and velocities.

V (m/s) | First (rad/s) | Second (rad/s) | Third (rad/s) | Fourth (rad/s) |
---|---|---|---|---|

0 | 165.46 | 501.67 | 992.99 | 1644.95 |

10 | 165.41 | 501.53 | 992.83 | 1644.78 |

30 | 165.01 | 500.42 | 991.51 | 1643.36 |

50 | 164.22 | 498.19 | 988.88 | 1640.52 |

100 | 160.65 | 487.64 | 976.43 | 1627.14 |

The first natural frequency and velocity relationship.

The input variables are listed in Table

The resonance failure probability of curved pipe conveying fluid at different velocity conditions.

Working condition | First failure probability | Second failure probability |
---|---|---|

v=0 m/s | 0.0071463 | 0.0231528 |

v=10 m/s | 0.0070448 | 0.0308182 |

v=20 m/s | 0.0062525 | 0.0373823 |

It can be seen from Table

Table

The sobol indices with different coefficients of variance.

| index | | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|---|

0.05 | | 0.6528 | 8.4508 | 6.6361 | 477.4792 | 6.1049 | 0.2442 | 8414.9 | 9.0357 | 475.8557 | 0.3755 |

| 0.4032 | 0.5625 | 0.2930 | 0.2784 | 0.3987 | 0.4590 | 0.9991 | 0.4626 | 0.8455 | 0.2553 | |

0.1 | | 0.0796 | 0.1262 | 0.00085 | 0.09031 | 0.00043 | 0 | 0.1038 | 0 | 0.0697 | 0 |

| 0.09796 | 0.38659 | 0.0319 | 0.20620 | 0.08094 | 0.0983 | 0.80845 | 0.0890 | 0.58185 | 0.09839 |

The Sobol indices of different coefficient of variance: main effect indices (a) and total effect indices (b).

From Figure

When the coefficient of variance is 0.1, the main effect indices change. Particularly the main index of out pipe diameter

Figure

The Sobol indices of different velocity: main effect indices (a) and total effect indices (b).

When the velocity increases from 0 m/s to 30 m/s, the main and total effect indices remain unchanged almost. The importance ranking under three velocity conditions still remains as

From Table

The resonance reliability GSA with different velocities

| index | | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|---|

0 | | 6.651E-6 | 8.603E-5 | 6.752 E-5 | 4.834E-3 | 6.211 E-5 | 2.494E-6 | 8.442E-2 | 9.195 E-5 | 4.8215E-3 | 3.817E-6 |

| 0.4028 | 0.5619 | 0.2926 | 0.2781 | 0.3981 | 0.4582 | 0.9991 | 0.4619 | 0.8452 | 0.2549 | |

10 | | 6.528e-6 | 8.450E-5 | 6.636e-5 | 4.774E-3 | 6.105E-5 | 2.442E-6 | 8.415E-2 | 9.035E-5 | 4.7586E-3 | 3.755E-6 |

| 0.40322 | 0.56258 | 0.29308 | 0.27849 | 0.39877 | 0.459 | 0.99913 | 0.46267 | 0.8456 | 0.25537 | |

30 | | 5.6376e-6 | 7.3446e-5 | 5.795e-5 | 0.0043393 | 5.3316e-5 | 2.0692e-6 | 0.078957 | 7.8782e-5 | 0.0043244 | 3.301E-6 |

| 0.40583 | 0.58354 | 0.29737 | 0.27726 | 0.4016 | 0.46314 | 0.99923 | 0.47664 | 0.84785 | 0.25557 |

Table

The main and total effect indices with different pressures.

| index | | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|---|

0 Mpa | | 0 | 1.935e-85 | 1.955E-85 | 2.385E-85 | 2.35E-85 | 1.35E-87 | 0.34995 | 5.88E-100 | 4.34E-85 | 0 |

| 0.1110 | 0.06435 | 0.18587 | 0.05004 | 0.111135 | 0.26033 | 1 | 0.11100 | 0.26248 | 0.11100 | |

10Mpa | | 6.528e-6 | 8.4508e-5 | 6.6361e-5 | 0.0047747 | 6.1049e-5 | 2.4422e-6 | 0.08414 | 9.0357e-5 | 0.004758 | 3.7551e-6 |

| 0.4032 | 0.5625 | 0.29307 | 0.2784 | 0.3987 | 0.4590 | 0.9991 | 0.4626 | 0.84559 | 0.25537 |

From Figure

The Sobol indices with different pressure: main effect indices (a) and total effect indices (b).

In this paper, the uncertainty of input variables on the natural frequency of the curved pipe conveying fluid is considered. The resonance failure probability and Sobol’s GSA are calculated based on TIS method. By decreasing the uncertainty of input variables with high main effect indices, the most reduction of failure probability can be obtained. By decreasing the uncertainty of the input variables with small total effect indices close to zero, the failure probability will not be reduced significantly. Different velocity and pressure working conditions are calculated, respectively.

According to the results of natural frequency, with the increasing of velocity, the natural frequency decreases slowly until the velocity

All data included in this study are available upon request by contact with the corresponding author.

The authors declare that they have no conflicts of interest.

The work was supported by the National Natural Science Foundation of China (Grant No. 51505388).