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SUNG Wen-pei, LIN Cheng-I, SHIH Ming-hsiang, GO Cheer-germ. Analysis modeling for plate buckling load of vibration test[J]. Journal of Zhejiang University Science A, 2005, 6(2): 132~140.

@article{title="Analysis modeling for plate buckling load of vibration test",

author="SUNG Wen-pei, LIN Cheng-I, SHIH Ming-hsiang, GO Cheer-germ",

journal="Journal of Zhejiang University Science A",

volume="6",

number="2",

pages="132~140",

year="2005",

publisher="Zhejiang University Press & Springer",

doi="10.1631/jzus.2005.A0132"

}

%0 Journal Article

%T Analysis modeling for plate buckling load of vibration test

%A SUNG Wen-pei

%A LIN Cheng-I

%A SHIH Ming-hsiang

%A GO Cheer-germ

%J Journal of Zhejiang University SCIENCE A

%V 6

%N 2

%P 132~140

%@ 1673-565X

%D 2005

%I Zhejiang University Press & Springer

%DOI 10.1631/jzus.2005.A0132

TY - JOUR

T1 - Analysis modeling for plate buckling load of vibration test

A1 - SUNG Wen-pei

A1 - LIN Cheng-I

A1 - SHIH Ming-hsiang

A1 - GO Cheer-germ

J0 - Journal of Zhejiang University Science A

VL - 6

IS - 2

SP - 132

EP - 140

%@ 1673-565X

Y1 - 2005

PB - Zhejiang University Press & Springer

ER -

DOI - 10.1631/jzus.2005.A0132

**Abstract: **In view of the recent technological development, the pursuit of safe high-precision structural designs has been the goal of most structural designers. To bridge the gap between the construction theories and the actual construction techniques, safety factors are adopted for designing the strength loading of structural members. If safety factors are too conservative, the extra building materials necessary will result in high construction cost. Thus, there has been a tendency in the construction field to derive a precise buckling load analysis model of member in order to establish accurate safety factors. A numerical analysis model, using modal analysis to acquire the dynamic function calculated by dynamic parameter to get the buckling load of member, is proposed in this paper. The fixed and simple supports around the circular plate are analyzed by this proposed method. And then, the monte Carlo method and the normal distribution method are used for random sampling and measuring errors of numerical simulation respectively. The analysis results indicated that this proposed method only needs to apply modal parameters of 7×7 test points to obtain a theoretical value of buckling load. Moreover, the analysis method of inequality-distant test points produces better analysis results than the other methods.

**
**

. INTRODUCTION

Experiments using dynamic theories include that of: (1) Lurie (

A dynamic analysis model is proposed to acquire buckling load of plate. We used the dynamic measured data from selected test points and by modal analysis got the modal parameters-mode shape and frequency; and then, derived a flexible matrix with the above model parameters. Force analysis was used to get the flexible matrix of equivalent force and the characteristic equation for determining the buckling load of the member. Under the equivalence, the buckling deformation function of the member was derived from the deformation tested at the test points and simulated by Lagrange’s Interpolation Equations. In this paper, we took the circular model member of ideal margins and fixed restraints on each side as an example. First, we used the given frequency and the corresponding mode shape to conduct the verification of the analysis model and then discussed the number and location of the test points and the mode shapes composing a linear isolated equation with the 1st, 2nd, 3rd, 4th, 5th, … values extracted from the vibration equation; and also discussed the impact of measured deviation caused by outside factors on buckling load. Monte Carlo Method was used to generate random scatter to simulate the values of measured deviations. Finally, we modeled the elements of the circular model member with simple restraint on each side based on the results of the above analysis, and used finite element analysis method to acquire modal parameters. Then, we verified the dynamic analysis model with the modal parameters obtained from measured data. The proposed analysis model is applicable to all kinds of loading and boundary conditions.

. ANALYSIS METHOD

Where,

Where:

Where, [

Where,

Solving the Eq.(

The theoretical model shows that [

Assuming deformation function is

Then substitute Eq.(

The forced vibration dynamic equation of circular plate is shown as follows (Meirovitch,

Assuming the deformation function is

Eq.(

Using the characteristics of Modal Orthogonal, Eq.(

Assuming the external force

The properties of Dirac Delta Function are shown as follows:

Substitution of Eqs.(

Substitution of initial conditions

Substitute Eq.(

If Ω→0, the external force approaches static load and the influence function can be shown as follows:

Use of the three-point method of Simpson’s rule yields the following expression for generalized mass

The used constant values of Eq.(

. VERIFICATION OF THEORETICAL MODEL

The normal random variable is produced through the uniform distribution random variation on the central limit theorem. The distribution situation is 99.7% of the weight factor plus/minus 100%. The standard deviation is 1/3. This diagram is shown in Fig.

Deflected rigidity D (kg(cm^{2}) |
Area of mass ( (kg/cm^{2}) |
Radius a (cm) |

1 | 1 | 1 |

The feasibility study of this proposed theoretical analysis model of buckling loads used the given theoretical formula of natural frequency and modal shape where the boundary conditions of circular plate are fixed. The analysis results are shown in Tables

Test point arrangement | No. of test points | Analysis solution | Theoretical solution | Error ratio |

Equidistant test point | 5(5 | 35.270 | 14.684 | +140.19 % |

7(7 | 14.895 | 14.684 | +1.4376 % | |

9(9 | 14.872 | 14.684 | +1.1127 % | |

Not equidistant test point 1 | 5(5 | 18.021 | 14.684 | +22.725 % |

7(7 | 14.879 | 14.684 | +1.3279 % | |

9(9 | 14.838 | 14.684 | +1.0488 % | |

Not equidistant test point 2 | 5(5 | 17.898 | 14.684 | +21.888 % |

7(7 | 14.869 | 14.684 | +1.2599 % | |

9(9 | 14.842 | 14.684 | +1.0760 % |

No. of test points | No. of modal shape | Analytical solution | Theoretical solution | Error ratio |

The first mode | 15.196 | 14.684 | 3.483 % | |

The second mode | 14.911 | 14.684 | 1.548 % | |

7(7 | The third mode | 14.911 | 14.684 | 1.544 % |

The forth mode | 14.902 | 14.684 | 1.485 % | |

The fifth mode | 14.895 | 14.684 | 1.438 % |

No. of test points | Aspect | Position |

5(5 | r direction |
0, a/4, a/2, 3a/4, a |

( direction |
0, (/2, (, 3(/2, 2( | |

7(7 | r direction |
0, a/6, a/3, a/2, 2a/3, 5a/6, a |

( direction |
0, (/3, 2(/3, (, 4(/3, 5(/3, 2( | |

9(9 | r direction |
0, a/8, a/4, 3a/8, a/2, 5a/8, 3a/4, 7a/8, a |

( direction |
0, (/4, (/2, 3(/4, (, 5(/4, 3(/2, 7(/4, 2( |

No. of test points | Aspect | Position |

5(5 | r direction |
0, a/5, a/2, 4a/5, a |

( direction |
0, (/2, (, 3(/2, 2( | |

7(7 | r direction |
0, a/8, a/4, a/2, 3a/4, 7a/8, a |

( direction |
0, (/3, 2(/3, (, 4(/3, 5(/3, 2( | |

9(9 | r direction |
0, a/10, a/5, 3a/10, a/2, 7a/10, 4a/5, 9a/10, a |

( direction |
0, (/4, (/2, 3(/4, (, 5(/4, 3(/2, 7(/4, 2( |

No. of test points | Aspect | Position |

5(5 | r direction |
0, a/5, 2a/5, 7a/10, a |

( direction |
0, (/2, (, 3(/2, 2( | |

7(7 | r direction |
0, a/8, a/4, 3a/8, 7a/12, 19a/24, a |

( direction |
0, (/3, 2(/3, (, 4(/3, 5(/3, 2( | |

9(9 | r direction |
0, a/10, a/5, 3a/10, 2a/5, 3a/5, 4a/5, 9a/10, a |

( direction |
0, (/4, (/2, 3(/4, (, 5(/4, 3(/2, 7(/4, 2( |

The theoretical formulas of natural frequency and modal shape of circular plate with fixed support are expressed as Eqs.(

Then, the circular plate with simple support is used as an example. According to the analysis results of Table

No. of test points | Style of test point | Analytical solution | Theoretical solution | Error ratio |

7(7 | Equidistant test point | 4.263 | 4.204 | 1.483% |

Not equidistant point | 4.248 | 4.204 | 1.047% |

. OUTCOME OF ERROR ANALYSIS

2% maximum measuring error | 4% maximum measuring error | 6% maximum measuring error | |||||||||

Error ratio (%) | Times | Error ratio (%) | Times | Error ratio (%) | Times | Error ratio (%) | Times | Error ratio (%) | Times | Error ratio (%) | Times |

(0.115 | 0 | (0.225 | 0 | 0.015 | 47 | (0.365 | 0 | (0.125 | 30 | 0.115 | 8 |

(0.105 | 1 | (0.215 | 2 | 0.025 | 34 | (0.355 | 1 | (0.115 | 25 | 0.125 | 5 |

(0.095 | 3 | (0.205 | 4 | 0.035 | 29 | (0.345 | 0 | (0.105 | 28 | 0.135 | 5 |

(0.085 | 13 | (0.195 | 5 | 0.045 | 30 | (0.335 | 1 | (0.095 | 30 | 0.145 | 4 |

(0.075 | 25 | (0.185 | 14 | 0.055 | 22 | (0.325 | 0 | (0.085 | 32 | 0.155 | 2 |

(0.065 | 38 | (0.175 | 18 | 0.065 | 18 | (0.315 | 3 | (0.075 | 26 | 0.165 | 3 |

(0.055 | 63 | (0.165 | 23 | 0.075 | 19 | (0.305 | 12 | (0.065 | 41 | 0.175 | 1 |

(0.045 | 59 | (0.155 | 39 | 0.085 | 12 | (0.295 | 7 | (0.055 | 36 | 0.185 | 0 |

(0.035 | 74 | (0.145 | 29 | 0.095 | 17 | (0.285 | 11 | (0.045 | 31 | 0.195 | 1 |

(0.025 | 74 | (0.135 | 31 | 0.105 | 8 | (0.275 | 19 | (0.035 | 38 | 0.205 | 0 |

(0.015 | 92 | (0.125 | 35 | 0.115 | 4 | (0.265 | 23 | (0.025 | 35 | 0.215 | 1 |

(0.005 | 99 | (0.115 | 31 | 0.125 | 4 | (0.255 | 25 | (0.015 | 27 | 0.225 | 0 |

0.005 | 96 | (0.105 | 28 | 0.135 | 3 | (0.245 | 22 | (0.005 | 28 | ||

0.015 | 113 | (0.095 | 44 | 0.145 | 1 | (0.235 | 21 | 0.005 | 30 | ||

0.025 | 74 | (0.085 | 29 | 0.155 | 1 | (0.225 | 18 | 0.015 | 28 | ||

0.035 | 62 | (0.075 | 35 | 0.165 | 0 | (0.215 | 22 | 0.025 | 20 | ||

0.045 | 46 | (0.065 | 39 | 0.175 | 1 | (0.205 | 30 | 0.035 | 19 | ||

0.055 | 39 | (0.055 | 42 | 0.185 | 0 | (0.195 | 26 | 0.045 | 16 | ||

0.065 | 13 | (0.045 | 58 | (0.185 | 19 | 0.055 | 20 | ||||

0.075 | 7 | (0.035 | 38 | (0.175 | 24 | 0.065 | 14 | ||||

0.085 | 5 | (0.025 | 50 | (0.165 | 22 | 0.075 | 8 | ||||

0.095 | 2 | (0.015 | 46 | (0.155 | 20 | 0.085 | 13 | ||||

0.105 | 2 | (0.005 | 61 | (0.145 | 21 | 0.095 | 12 | ||||

0.115 | 0 | 0.005 | 49 | (0.135 | 27 | 0.105 | 9 |

8% maximum measuring error | 10% maximum measuring error | ||||||||||

Error ratio (%) | Times | Error ratio (%) | Times | Error ratio (%) | Times | Error ratio (%) | Times | Error ratio (%) | Times | Error ratio (%) | Times |

(0.485 | 0 | (0.205 | 19 | 0.075 | 5 | (0.585 | 0 | (0.305 | 15 | (0.025 | 16 |

(0.475 | 1 | (0.195 | 20 | 0.085 | 5 | (0.575 | 1 | (0.295 | 13 | (0.015 | 19 |

(0.465 | 1 | (0.185 | 29 | 0.095 | 7 | (0.565 | 1 | (0.285 | 13 | (0.005 | 14 |

(0.455 | 0 | (0.175 | 17 | 0.105 | 5 | (0.555 | 1 | (0.275 | 23 | 0.005 | 13 |

(0.445 | 1 | (0.165 | 21 | 0.115 | 9 | (0.545 | 0 | (0.265 | 16 | 0.015 | 6 |

(0.435 | 2 | (0.155 | 26 | 0.125 | 3 | (0.535 | 3 | (0.255 | 15 | 0.025 | 9 |

(0.425 | 5 | (0.145 | 24 | 0.135 | 4 | (0.525 | 5 | (0.245 | 19 | 0.035 | 9 |

(0.415 | 5 | (0.135 | 19 | 0.145 | 3 | (0.515 | 2 | (0.235 | 28 | 0.045 | 8 |

(0.405 | 5 | (0.125 | 28 | 0.155 | 4 | (0.505 | 4 | (0.225 | 13 | 0.055 | 5 |

(0.395 | 9 | (0.115 | 24 | 0.165 | 3 | (0.495 | 6 | (0.215 | 20 | 0.065 | 5 |

(0.385 | 4 | (0.105 | 30 | 0.175 | 2 | (0.485 | 4 | (0.205 | 25 | 0.075 | 5 |

(0.375 | 18 | (0.095 | 28 | 0.185 | 1 | (0.475 | 15 | (0.195 | 19 | 0.085 | 4 |

(0.365 | 14 | (0.085 | 28 | 0.195 | 2 | (0.465 | 12 | (0.185 | 23 | 0.095 | 3 |

(0.355 | 20 | (0.075 | 23 | 0.205 | 0 | (0.455 | 6 | (0.175 | 20 | 0.105 | 3 |

(0.345 | 16 | (0.065 | 26 | 0.215 | 1 | (0.445 | 14 | (0.165 | 24 | 0.115 | 4 |

(0.335 | 21 | (0.055 | 29 | 0.225 | 0 | (0.435 | 12 | (0.155 | 21 | 0.125 | 6 |

(0.325 | 17 | (0.045 | 21 | 0.235 | 1 | (0.425 | 19 | (0.145 | 21 | 0.135 | 2 |

(0.315 | 25 | (0.035 | 22 | 0.245 | 1 | (0.415 | 17 | (0.135 | 23 | 0.145 | 4 |

(0.305 | 13 | (0.025 | 19 | 0.255 | 0 | (0.405 | 16 | (0.125 | 28 | 0.155 | 1 |

. CONCLUSION

Take a circular plate using fixed cyclic support as an example for using this proposed analysis model to analyze its buckling load with equidistant and not equidistant test points. The analysis results revealed that this proposed method only needs to apply the modal parameters of 7×7 test points for acquiring the theoretical approximate values of buckling load. Moreover, the analysis method using not equidistant test points produces better analysis results, mainly because that the test points of this method gather around the inflection point of the deformation curve.

The results of error simulation experiment indicated (1) the error range of the buckling load enlarges with the increase of the maximum measured error; (2) in order to obtain more exact analysis solution of the buckling load with minimum error, doing the calculation several times and getting the mean value of them is suggested.

* Project supported by the National Science Council of Taiwan (No. NSC 93-2211-E-167-002), and Wu-Feng Institute of Technology, Taiwan, China

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