Research Papers: Design and Analysis

Equilibrium Based Curve Fitting Method for Test Data With Nonuniform Variance

[+] Author and Article Information
Zhigang Wei

Tenneco, Inc.,
Grass Lake, MI 49240
e-mail: zwei@tenneco.com

Shervin Maleki

The Welding Institute, Ltd.,
Granta Park,
Great Abington, Cambridge CB21 6AL, UK

Kamran Nikbin

Department of Mechanical Engineering,
Imperial College,
South Kensington Campus,
London SW7 2AZ, UK

1Corresponding author.

Contributed by the Pressure Vessel and Piping Division of ASME for publication in the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received May 13, 2012; final manuscript received August 18, 2013; published online January 29, 2014. Assoc. Editor: Albert E. Segall.

J. Pressure Vessel Technol 136(2), 021203 (Jan 29, 2014) (7 pages) Paper No: PVT-12-1058; doi: 10.1115/1.4026020 History: Received May 13, 2012; Revised August 18, 2013

A recently developed equilibrium based curve/surface fitting method is extended to linear function with heteroscedastic data (variable variance). The concept of equilibrium weighted “force” and “moment” is proposed to derive curve fitting formulae, which are exactly the same as that obtained with the traditional weighted least squares (LS) method for a linear function. Furthermore, a system of four equations, i.e., a “force” equilibrium equation, a “moment” equilibrium equation, an “equivalency” equation, and a “moment balance” equation have been established to solve both mean curve and the standard deviation simultaneously. Finally, the application of these methods to data of fatigue and creep lives is presented.

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Fig. 1

Ideal curve fitting methods for given data patterns: (a) vertical offsets method and (b) horizontal offsets method

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Fig. 2

Equilibrium re-establishment of (a) horizontal patterned data with two added data blocks and (b) horizontal patterned data with added antisymmetrical data blocks

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Fig. 3

360 deg closed welds data and the fit curves with standard horizontal offsets method (dashed–dotted line) and vertical method (solid line)

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Fig. 4

(a) a data set with heteroscedastic characteristics [1], and plots of residuals against x with (b) w(xi) = 1; (c) w(xi) = 1/xi; (d) w(xi) = 1/xi2; and (e) w(xi) = 1/xi1/2

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Fig. 5

The schematic of the equilibrium establishment of (a) the original data with heteroscedasticity and (b) transformed data with a weight function

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Fig. 6

The equilibrium mechanism of the two-stress level heteroscedastic data

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Fig. 7

The mean and the design curves obtained with the equilibrium mechanism for two-stress level heteroscedastic fatigue data.

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Fig. 8

The mean curves obtained for the heteroscedastic creep data (a) without trimming operation and (b) with trimming operation

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Fig. 9

Plots of residuals against x for (a) without weight and (b) with the introduction of weight

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Fig. 10

The mean and the design curves obtained with the equilibrium mechanism for multiple-stress level heteroscedastic creep growth rate data




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