Statistical Background - Revision history

Lisa Hacker at 18:56, 15 September 2023

2023-09-15T18:56:40Z

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	For example, when considering the Weibull distribution, the scale parameter, <math>\eta \,\!</math>, is chosen to be the life characteristic that is stress-dependent while <math>\beta \,\!</math> is assumed to remain constant across different stress levels. A life-stress relationship is then assigned to <math>\eta .\,\!</math>		For example, when considering the Weibull distribution, the scale parameter, <math>\eta \,\!</math>, is chosen to be the life characteristic that is stress-dependent while <math>\beta \,\!</math> is assumed to remain constant across different stress levels. A life-stress relationship is then assigned to <math>\eta .\,\!</math>

	For a detailed discussion of this topic, see ~~[[Accelerated_Life_Testing_Data_Analysis_Reference\|~~ReliaSoft's Accelerated Life Testing Data Analysis Reference]].		For a detailed discussion of this topic, see ReliaSoft's Accelerated Life Testing Data Analysis Reference.

Richard House: /* A Brief Introduction to Life-Stress Relationships */

2016-01-05T17:13:58Z

A Brief Introduction to Life-Stress Relationships

← Older revision		Revision as of 17:13, 5 January 2016
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	Traditionally in a reliability block diagram, one assumes independence and thus an item's failure characteristics can be fully described by its failure distribution. However, if the configuration includes load sharing redundancy, then a single failure distribution is no longer sufficient to describe an item's failure characteristics. Instead, the item will fail differently when operating under different loads and the load applied to the component will vary depending on the performance of the other component(s) in the configuration. Therefore, a more complex model is needed to fully describe the failure characteristics of such blocks. This model must describe both the effect of the load (or stress) on the life of the product and the probability of failure of the item at the specified load. The models, theory and methodology used in Quantitative Accelerated Life Testing (QALT) data analysis can be used to obtain the desired model for this situation. The objective of QALT analysis is to relate the applied stress to life (or a life distribution). Identically in the load sharing case, one again wants to relate the applied stress (or load) to life. The following figure graphically illustrates the probability density function (''pdf'') for a standard item, where only a single distribution is required.		Traditionally in a reliability block diagram, one assumes independence and thus an item's failure characteristics can be fully described by its failure distribution. However, if the configuration includes load sharing redundancy, then a single failure distribution is no longer sufficient to describe an item's failure characteristics. Instead, the item will fail differently when operating under different loads and the load applied to the component will vary depending on the performance of the other component(s) in the configuration. Therefore, a more complex model is needed to fully describe the failure characteristics of such blocks. This model must describe both the effect of the load (or stress) on the life of the product and the probability of failure of the item at the specified load. The models, theory and methodology used in Quantitative Accelerated Life Testing (QALT) data analysis can be used to obtain the desired model for this situation. The objective of QALT analysis is to relate the applied stress to life (or a life distribution). Identically in the load sharing case, one again wants to relate the applied stress (or load) to life. The following figure graphically illustrates the probability density function (''pdf'') for a standard item, where only a single distribution is required.

	[[Image:chp3LSR1.png\|center\|300px\|Single ''pdf'']]		[[Image:chp3LSR1.png\|center\|300px\|Single ''pdf''\|link=]]
	The next figure represents a load sharing item by using a 3-D surface that illustrates the ''pdf'' , load and time.		The next figure represents a load sharing item by using a 3-D surface that illustrates the ''pdf'' , load and time.
	[[Image:Pdf_lifestress1.png\|center\|300px\|''pdf'' and life-stress relationship.]]		[[Image:Pdf_lifestress1.png\|center\|300px\|''pdf'' and life-stress relationship.\|link=]]
	The following figure shows the reliability curve for a load sharing item vs. the applied load.		The following figure shows the reliability curve for a load sharing item vs. the applied load.
	[[Image:chp3reliabilityvsloadsurface.png\|center\|~~300px~~\|Reliability and life-stress relationship.]]		[[Image:chp3reliabilityvsloadsurface.png\|center\|400px\|Reliability and life-stress relationship.\|link=]]

	To formulate the model, a life distribution is combined with a life-stress relationship. The distribution choice is based on the product's failure characteristics while the life-stress relationship is based on how the stress affects the life characteristics. The following figure graphically shows these elements of the formulation.		To formulate the model, a life distribution is combined with a life-stress relationship. The distribution choice is based on the product's failure characteristics while the life-stress relationship is based on how the stress affects the life characteristics. The following figure graphically shows these elements of the formulation.
	[[Image:chp3formulation.png\|center\|550px\|A life distribution and a life-stress relationship.]]		[[Image:chp3formulation.png\|center\|550px\|A life distribution and a life-stress relationship.\|link=]]
	The next figure shows the combination of both an underlying distribution and a life-stress model by plotting a ''pdf'' against both time and stress.		The next figure shows the combination of both an underlying distribution and a life-stress model by plotting a ''pdf'' against both time and stress.
	[[Image:chp3formulation2.png\|center\|400px\|''pdf'' vs. time and stress.]]		[[Image:chp3formulation2.png\|center\|400px\|''pdf'' vs. time and stress.\|link=]]

	The assumed underlying life distribution can be any life distribution. The most commonly used life distributions include the Weibull, the exponential and the lognormal. The life-stress relationship describes how a specific life characteristic changes with the application of stress. The life characteristic can be any life measure such as the mean, median, <math>R(x)\,\!</math>, <math>F(x)\,\!</math>, etc. It is expressed as a function of stress. Depending on the assumed underlying life distribution, different life characteristics are considered. Typical life characteristics for some distributions are shown in the next table.		The assumed underlying life distribution can be any life distribution. The most commonly used life distributions include the Weibull, the exponential and the lognormal. The life-stress relationship describes how a specific life characteristic changes with the application of stress. The life characteristic can be any life measure such as the mean, median, <math>R(x)\,\!</math>, <math>F(x)\,\!</math>, etc. It is expressed as a function of stress. Depending on the assumed underlying life distribution, different life characteristics are considered. Typical life characteristics for some distributions are shown in the next table.

Richard House: /* Example 2 */

2015-12-11T18:42:37Z

Example 2

← Older revision		Revision as of 18:42, 11 December 2015
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	Consider a system with a load being held in place by two rigid members, as shown next.		Consider a system with a load being held in place by two rigid members, as shown next.

	[[Image:chp3ex2.png\|center\|350px\|System for Example 2.]]		[[Image:chp3ex2.png\|center\|350px\|System for Example 2.\|link=]]

	:• Let <math>A=\,\!</math> event of failure of Component 1.		:• Let <math>A=\,\!</math> event of failure of Component 1.

Richard House: /* Example 1 */

2015-12-11T18:42:27Z

Example 1

← Older revision		Revision as of 18:42, 11 December 2015
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	Consider a system where two hinged members are holding a load in place, as shown next.		Consider a system where two hinged members are holding a load in place, as shown next.

	[[Image:chp3ex1.png\|center\|350px\|System for Example 1]]		[[Image:chp3ex1.png\|center\|350px\|System for Example 1\|link=]]

	The system fails if either member fails and the load is moved from its position.		The system fails if either member fails and the load is moved from its position.

Richard House: /* Example 2 */

2015-12-11T18:42:02Z

Example 2

← Older revision		Revision as of 18:42, 11 December 2015
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	Consider a system with a load being held in place by two rigid members, as shown next.		Consider a system with a load being held in place by two rigid members, as shown next.

	[[Image:chp3ex2.png\|center\|~~250px~~\|System for Example 2.]]		[[Image:chp3ex2.png\|center\|350px\|System for Example 2.]]

	:• Let <math>A=\,\!</math> event of failure of Component 1.		:• Let <math>A=\,\!</math> event of failure of Component 1.

Richard House: /* Example 1 */

2015-12-11T18:41:52Z

Example 1

← Older revision		Revision as of 18:41, 11 December 2015
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	Consider a system where two hinged members are holding a load in place, as shown next.		Consider a system where two hinged members are holding a load in place, as shown next.

	[[Image:chp3ex1.png\|center\|~~250px~~\|System for Example 1]]		[[Image:chp3ex1.png\|center\|350px\|System for Example 1]]

	The system fails if either member fails and the load is moved from its position.		The system fails if either member fails and the load is moved from its position.

Richard House at 20:21, 17 September 2012

2012-09-17T20:21:10Z

Show changes

Melinda Caroline: /* A Brief Introduction to Life-Stress Relationships */

2012-09-17T17:52:50Z

A Brief Introduction to Life-Stress Relationships

← Older revision		Revision as of 17:52, 17 September 2012
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	[[Image:chp3LSR1.png\|center\|300px\|Single ''pdf'']]		[[Image:chp3LSR1.png\|center\|300px\|Single ''pdf'']]
	The next figure represents a load sharing item by using a 3-D surface that illustrates the ''pdf'' , load and time.		The next figure represents a load sharing item by using a 3-D surface that illustrates the ''pdf'' , load and time.
	[[Image:~~pdfnlifestress~~.png\|center\|300px\|''pdf'' and life-stress relationship.]]		[[Image:Pdf_lifestress1.png\|center\|300px\|''pdf'' and life-stress relationship.]]
	The following figure shows the reliability curve for a load sharing item vs. the applied load.		The following figure shows the reliability curve for a load sharing item vs. the applied load.
	[[Image:chp3reliabilityvsloadsurface.png\|center\|300px\|Reliability and life-stress relationship.]]		[[Image:chp3reliabilityvsloadsurface.png\|center\|300px\|Reliability and life-stress relationship.]]

Nicolette Young: /* A Brief Introduction to Life-Stress Relationships */

2012-09-14T23:30:11Z

A Brief Introduction to Life-Stress Relationships

← Older revision		Revision as of 23:30, 14 September 2012
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	[[Image:chp3LSR1.png\|center\|300px\|Single ''pdf'']]		[[Image:chp3LSR1.png\|center\|300px\|Single ''pdf'']]
	The next figure represents a load sharing item by using a 3-D surface that illustrates the ''pdf'' , load and time.		The next figure represents a load sharing item by using a 3-D surface that illustrates the ''pdf'' , load and time.
	[[Image:pdfnlifestress.png\|center\|300px\|''pdf'' and life-stress relationship]]		[[Image:pdfnlifestress.png\|center\|300px\|''pdf'' and life-stress relationship.]]
	The following figure shows the reliability curve for a load sharing item vs. the applied load.		The following figure shows the reliability curve for a load sharing item vs. the applied load.
	[[Image:chp3reliabilityvsloadsurface.png\|center\|300px\|Reliability and life-stress relationship.]]		[[Image:chp3reliabilityvsloadsurface.png\|center\|300px\|Reliability and life-stress relationship.]]

Nicolette Young: /* A Brief Introduction to Life-Stress Relationships */

2012-09-14T23:29:01Z

A Brief Introduction to Life-Stress Relationships

← Older revision		Revision as of 23:29, 14 September 2012
Line 170:		Line 170:
	[[Image:chp3LSR1.png\|center\|300px\|Single ''pdf'']]		[[Image:chp3LSR1.png\|center\|300px\|Single ''pdf'']]
	The next figure represents a load sharing item by using a 3-D surface that illustrates the ''pdf'' , load and time.		The next figure represents a load sharing item by using a 3-D surface that illustrates the ''pdf'' , load and time.
	[[Image:pdfnlifestress.png\|center\|300px\|''pdf'' and life-stress relationship~~.center~~]]		[[Image:pdfnlifestress.png\|center\|300px\|''pdf'' and life-stress relationship]]
	The following figure shows the reliability curve for a load sharing item vs. the applied load.		The following figure shows the reliability curve for a load sharing item vs. the applied load.
	[[Image:chp3reliabilityvsloadsurface.png\|center\|300px\|Reliability and life-stress relationship.]]		[[Image:chp3reliabilityvsloadsurface.png\|center\|300px\|Reliability and life-stress relationship.]]