Stress-Strength Analysis - Revision history

Lisa Hacker at 19:11, 15 September 2023

2023-09-15T19:11:31Z

← Older revision		Revision as of 19:11, 15 September 2023
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	{{Template:~~LDABOOK_SUB~~\|Additional Reliability Analysis Tools\|Stress-Strength Analysis}}		{{Template:LDABOOK\|Additional Reliability Analysis Tools\|Stress-Strength Analysis}}

	Stress-strength analysis has been used in mechanical component design. The probability of failure is based on the probability of stress exceeding strength. The following equation is used to calculate the expected probability of failure, ''F'':		Stress-strength analysis has been used in mechanical component design. The probability of failure is based on the probability of stress exceeding strength. The following equation is used to calculate the expected probability of failure, ''F'':

Kate Racaza at 23:25, 12 May 2014

2014-05-12T23:25:26Z

← Older revision		Revision as of 23:25, 12 May 2014
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	::<math>R=P[Stress\le Strength]=\int_{0}^{\infty }{{{f}_{Stress}}(x)\cdot {{R}_{Strength}}(x)}dx\,\!</math>		::<math>R=P[Stress\le Strength]=\int_{0}^{\infty }{{{f}_{Stress}}(x)\cdot {{R}_{Strength}}(x)}dx\,\!</math>

	The above ~~calculation assumes~~ both stress and strength are in the positive domain. For general cases, the expected reliability can be calculated using the following equation:		The equations above assume that both stress and strength are in the positive domain. For general cases, the expected reliability can be calculated using the following equation:

	::<math>R=P[{{X}_{1}}\le {{X}_{2}}]=\frac{1}{{{F}_{1}}(U)-{{F}_{1}}(L)}\int_{L}^{U}{{{f}_{1}}(x)\cdot {{R}_{2}}(x)}dx\,\!</math>		::<math>R=P[{{X}_{1}}\le {{X}_{2}}]=\frac{1}{{{F}_{1}}(U)-{{F}_{1}}(L)}\int_{L}^{U}{{{f}_{1}}(x)\cdot {{R}_{2}}(x)}dx\,\!</math>
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	\end{align}\,\!</math>		\end{align}\,\!</math>
	<br>		<br>
	When U = ~~infinite~~ and L = 0, the ~~above two equations are~~ the ~~same~~.		When U = infinity and L = 0, this equation becomes equal to previous equation (i.e., the equation for the expected reliability ''R'').

	==Confidence Intervals on the Probability==		==Confidence Intervals on the Probability==

Harry Guo: /* Variation in Model Parameters */

2013-11-25T23:05:24Z

Variation in Model Parameters

← Older revision		Revision as of 23:05, 25 November 2013
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	===Variation in Model Parameters===		===Variation in Model Parameters===
	If both the stress and strength distributions are estimated from data sets, then there are uncertainties associated with the estimated distribution parameters. These uncertainties will cause some degree of variation of the probability calculated from the stress-strength analysis. Therefore, we can use these uncertainties to estimate the confidence intervals on the calculated probability. To get the confidence intervals, we first calculate the variance of the reliability, <math>R\,\!</math>, ~~using~~:		If both the stress and strength distributions are estimated from data sets, then there are uncertainties associated with the estimated distribution parameters. These uncertainties will cause some degree of variation of the probability calculated from the stress-strength analysis. Therefore, we can use these uncertainties to estimate the confidence intervals on the calculated probability. To get the confidence intervals, we first calculate the variance of the reliability based on Taylor expansion by ignoring the 2nd order term. <math>R\,\!</math>, The approximation for the variance is:

	::<math>Var\left[ R \right]=\int_{0}^{\infty }{Var\left[ {{f}_{1}}(x) \right]}{{\left[ {{R}_{2}}(x) \right]}^{2}}dx+{{\int_{0}^{\infty }{\left[ {{f}_{1}}(x) \right]}}^{2}}Var\left[ {{R}_{2}}(x) \right]dx\,\!</math>		::<math>Var\left[ R \right]=\int_{0}^{\infty }{Var\left[ {{f}_{1}}(x) \right]}{{\left[ {{R}_{2}}(x) \right]}^{2}}dx+{{\int_{0}^{\infty }{\left[ {{f}_{1}}(x) \right]}}^{2}}Var\left[ {{R}_{2}}(x) \right]dx\,\!</math>

Richard House at 19:42, 26 September 2012

2012-09-26T19:42:16Z

Richard House: /* Variation in Probability Values */

2012-08-24T04:56:54Z

Variation in Probability Values

← Older revision		Revision as of 04:56, 24 August 2012
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	This is the reliability when the stress value is 567 and when the strength distribution is given. If stress is not a fixed value (i.e., it follows a distribution instead), then it can take values other than 567. For instance, if stress takes a value of 700, then we get another reliability value of <math>{R}(700)\,\!</math>. Since stress is a random variable, for any stress value <math>{x}_{i}\,\!</math>, there is a reliability value of		This is the reliability when the stress value is 567 and when the strength distribution is given. If stress is not a fixed value (i.e., it follows a distribution instead), then it can take values other than 567. For instance, if stress takes a value of 700, then we get another reliability value of <math>{R}(700)\,\!</math>. Since stress is a random variable, for any stress value <math>{x}_{i}\,\!</math>, there is a reliability value of
	<math>R({{x}_{i}})</math> calculated from the strength distribution. We will end up with many <math>R({{x}_{i}})</math>s or <math>R_{2}({{x}_{i}})\,\!</math>s. From these <math>R({{x}_{i}})\,\!</math>s, we can get the mean and variance of the reliability. In fact, its mean is the result from:		<math>R({{x}_{i}})\,\!</math> calculated from the strength distribution. We will end up with many <math>R({{x}_{i}})\,\!</math>s or <math>R_{2}({{x}_{i}})\,\!</math>s. From these <math>R({{x}_{i}})\,\!</math>s, we can get the mean and variance of the reliability. In fact, its mean is the result from:

	::<math>R=P[Stress\le Strength]=\int_{0}^{\infty }{{{f}_{Stress}}(x)\cdot {{R}_{Strength}}(x)}dx</math>		::<math>R=P[Stress\le Strength]=\int_{0}^{\infty }{{{f}_{Stress}}(x)\cdot {{R}_{Strength}}(x)}dx</math>

Richard House: /* Variation in Probability Values */

2012-08-24T04:56:03Z

Variation in Probability Values

← Older revision		Revision as of 04:56, 24 August 2012
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	::<math>R(567)=\Pr (Strength>567)={{R}_{2}}(567)</math>		::<math>R(567)=\Pr (Strength>567)={{R}_{2}}(567)</math>

	This is the reliability when the stress value is 567 and when the strength distribution is given. If stress is not a fixed value (i.e., it follows a distribution instead), then it can take values other than 567. For instance, if stress takes a value of 700, then we get another reliability value of <math>{R}(700)</math>. Since stress is a random variable, for any stress value <math>{x}_{i}</math>, there is a reliability value of		This is the reliability when the stress value is 567 and when the strength distribution is given. If stress is not a fixed value (i.e., it follows a distribution instead), then it can take values other than 567. For instance, if stress takes a value of 700, then we get another reliability value of <math>{R}(700)\,\!</math>. Since stress is a random variable, for any stress value <math>{x}_{i}\,\!</math>, there is a reliability value of
	<math>R({{x}_{i}})</math> calculated from the strength distribution. We will end up with many <math>R({{x}_{i}})</math>s or <math>R_{2}({{x}_{i}})</math>s. From these <math>R({{x}_{i}})</math>s, we can get the mean and variance of the reliability. In fact, its mean is the result from:		<math>R({{x}_{i}})</math> calculated from the strength distribution. We will end up with many <math>R({{x}_{i}})</math>s or <math>R_{2}({{x}_{i}})\,\!</math>s. From these <math>R({{x}_{i}})\,\!</math>s, we can get the mean and variance of the reliability. In fact, its mean is the result from:

	::<math>R=P[Stress\le Strength]=\int_{0}^{\infty }{{{f}_{Stress}}(x)\cdot {{R}_{Strength}}(x)}dx</math>		::<math>R=P[Stress\le Strength]=\int_{0}^{\infty }{{{f}_{Stress}}(x)\cdot {{R}_{Strength}}(x)}dx</math>

Richard House: /* Stress-Strength Analysis in Design for Reliability */

2012-08-24T04:55:03Z

Stress-Strength Analysis in Design for Reliability

← Older revision		Revision as of 04:55, 24 August 2012
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	The stress distribution is usually estimated from customer usage data, such as the mileage per year of a passenger car or the load distribution for a beam. The strength distribution, on the other hand, is affected by the material used in the component, the geometric dimensions and the manufacturing process.		The stress distribution is usually estimated from customer usage data, such as the mileage per year of a passenger car or the load distribution for a beam. The strength distribution, on the other hand, is affected by the material used in the component, the geometric dimensions and the manufacturing process.

	Because the stress distribution can be estimated from customer usage data, we can assume that <math>{f}_{Stress} </math> is known. Therefore, for a given reliability goal, the strength distribution <math> {R}_{Strength}</math> is the only unknown in the given equation. The factors that affect the strength distribution can be adjusted to obtain a strength distribution that meets the reliability goal. The following example shows how to use the Target Reliability Parameter Estimator tool in the stress-strength folio to obtain the parameters for a strength distribution that will meet a specified reliability goal.		Because the stress distribution can be estimated from customer usage data, we can assume that <math>{f}_{Stress} \,\!</math> is known. Therefore, for a given reliability goal, the strength distribution <math> {R}_{Strength}\,\!</math> is the only unknown in the given equation. The factors that affect the strength distribution can be adjusted to obtain a strength distribution that meets the reliability goal. The following example shows how to use the Target Reliability Parameter Estimator tool in the stress-strength folio to obtain the parameters for a strength distribution that will meet a specified reliability goal.

	===Example 2===		===Example 2===
	{{:Stress-Strength Analysis in Design for Reliability}}		{{:Stress-Strength Analysis in Design for Reliability}}

Richard House: /* Variation in Model Parameters */

2012-08-24T04:54:26Z

Variation in Model Parameters

← Older revision		Revision as of 04:54, 24 August 2012
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	===Variation in Model Parameters===		===Variation in Model Parameters===
	If both the stress and strength distributions are estimated from data sets, then there are uncertainties associated with the estimated distribution parameters. These uncertainties will cause some degree of variation of the probability calculated from the stress-strength analysis. Therefore, we can use these uncertainties to estimate the confidence intervals on the calculated probability. To get the confidence intervals, we first calculate the variance of the reliability, <math>R</math>, using:		If both the stress and strength distributions are estimated from data sets, then there are uncertainties associated with the estimated distribution parameters. These uncertainties will cause some degree of variation of the probability calculated from the stress-strength analysis. Therefore, we can use these uncertainties to estimate the confidence intervals on the calculated probability. To get the confidence intervals, we first calculate the variance of the reliability, <math>R\,\!</math>, using:

	::<math>Var\left[ R \right]=\int_{0}^{\infty }{Var\left[ {{f}_{1}}(x) \right]}{{\left[ {{R}_{2}}(x) \right]}^{2}}dx+{{\int_{0}^{\infty }{\left[ {{f}_{1}}(x) \right]}}^{2}}Var\left[ {{R}_{2}}(x) \right]dx</math>		::<math>Var\left[ R \right]=\int_{0}^{\infty }{Var\left[ {{f}_{1}}(x) \right]}{{\left[ {{R}_{2}}(x) \right]}^{2}}dx+{{\int_{0}^{\infty }{\left[ {{f}_{1}}(x) \right]}}^{2}}Var\left[ {{R}_{2}}(x) \right]dx</math>

	Variance of <math>{{f}_{1}}(x)</math> and <math>{{R}_{2}}(x)</math> can be estimated from the Fisher Information Matrix. For details, please see [[Confidence Bounds]].		Variance of <math>{{f}_{1}}(x)\,\!</math> and <math>{{R}_{2}}(x)\,\!</math> can be estimated from the Fisher Information Matrix. For details, please see [[Confidence Bounds]].

	Once the variance of the expected reliability is obtained, the two-sided confidence intervals can be calculated using:		Once the variance of the expected reliability is obtained, the two-sided confidence intervals can be calculated using:
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	:: <math>{{Z}_{1-\alpha /2}}</math> is the <math>1-\alpha/2</math> percentile of a standard normal distribution.		:: <math>{{Z}_{1-\alpha /2}}</math> is the <math>1-\alpha/2</math> percentile of a standard normal distribution.

	If the upper bound (U) and lower bound (L) are not infinite and 0, respectively, then the calculated variance of <math> R </math> is adjusted by <math>{{\left[ {1}/{\left( {{F}_{1}}(U)-{{F}_{1}}(L) \right)}\; \right]}^{2}}</math>.		If the upper bound (U) and lower bound (L) are not infinite and 0, respectively, then the calculated variance of <math> R \,\!</math> is adjusted by <math>{{\left[ {1}/{\left( {{F}_{1}}(U)-{{F}_{1}}(L) \right)}\; \right]}^{2}}\,\!</math>.

	===Variation in Probability Values===		===Variation in Probability Values===

Kate Racaza: /* Stress-Strength Analysis in Design for Reliability */

2012-08-13T02:32:56Z

Stress-Strength Analysis in Design for Reliability

← Older revision		Revision as of 02:32, 13 August 2012
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	==Stress-Strength Analysis in Design for Reliability==~~<!-- THIS SECTION HEADER IS LINKED TO EXTERNAL SEARCH IN THE W++/A HELP FILE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->~~		==Stress-Strength Analysis in Design for Reliability==
	As we know, the expected reliability is called from the following stress-strength calculation:		As we know, the expected reliability is called from the following stress-strength calculation:

Kate Racaza: /* Stress-Strength Analysis in Design for Reliability */ added editorial note about the header

2012-08-13T02:31:28Z

Stress-Strength Analysis in Design for Reliability: added editorial note about the header

← Older revision		Revision as of 02:31, 13 August 2012
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	==Stress-Strength Analysis in Design for Reliability==		==Stress-Strength Analysis in Design for Reliability==<!-- THIS SECTION HEADER IS LINKED TO EXTERNAL SEARCH IN THE W++/A HELP FILE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->

	As we know, the expected reliability is called from the following stress-strength calculation:		As we know, the expected reliability is called from the following stress-strength calculation:

← Older revision		Revision as of 19:42, 26 September 2012
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	Stress-strength analysis has been used in mechanical component design. The probability of failure is based on the probability of stress exceeding strength. The following equation is used to calculate the expected probability of failure, ''F'':		Stress-strength analysis has been used in mechanical component design. The probability of failure is based on the probability of stress exceeding strength. The following equation is used to calculate the expected probability of failure, ''F'':

	::<math>F=P[Stress\ge Strength]=\int_{0}^{\infty }{{{f}_{Strength}}(x)\cdot {{R}_{Stress}}(x)}dx</math>		::<math>F=P[Stress\ge Strength]=\int_{0}^{\infty }{{{f}_{Strength}}(x)\cdot {{R}_{Stress}}(x)}dx\,\!</math>

	The expected probability of success or the expected reliability, ''R'' , is calculated as:		The expected probability of success or the expected reliability, ''R'' , is calculated as:

	::<math>R=P[Stress\le Strength]=\int_{0}^{\infty }{{{f}_{Stress}}(x)\cdot {{R}_{Strength}}(x)}dx</math>		::<math>R=P[Stress\le Strength]=\int_{0}^{\infty }{{{f}_{Stress}}(x)\cdot {{R}_{Strength}}(x)}dx\,\!</math>

	The above calculation assumes both stress and strength are in the positive domain. For general cases, the expected reliability can be calculated using the following equation:		The above calculation assumes both stress and strength are in the positive domain. For general cases, the expected reliability can be calculated using the following equation:

	::<math>R=P[{{X}_{1}}\le {{X}_{2}}]=\frac{1}{{{F}_{1}}(U)-{{F}_{1}}(L)}\int_{L}^{U}{{{f}_{1}}(x)\cdot {{R}_{2}}(x)}dx</math>		::<math>R=P[{{X}_{1}}\le {{X}_{2}}]=\frac{1}{{{F}_{1}}(U)-{{F}_{1}}(L)}\int_{L}^{U}{{{f}_{1}}(x)\cdot {{R}_{2}}(x)}dx\,\!</math>

	where:		where:
	:: <math>L\le {{X}_{1}}\le U</math>		:: <math>L\le {{X}_{1}}\le U\,\!</math>
	::<math>\begin{align}		::<math>\begin{align}
	& {{X}_{1}}:\text{ Stress } \\		& {{X}_{1}}:\text{ Stress } \\
	& {{X}_{2}}:\text{ Strength } \\		& {{X}_{2}}:\text{ Strength } \\
	\end{align}</math>		\end{align}\,\!</math>
	<br>		<br>
	When U = infinite and L = 0, the above two equations are the same.		When U = infinite and L = 0, the above two equations are the same.
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	If both the stress and strength distributions are estimated from data sets, then there are uncertainties associated with the estimated distribution parameters. These uncertainties will cause some degree of variation of the probability calculated from the stress-strength analysis. Therefore, we can use these uncertainties to estimate the confidence intervals on the calculated probability. To get the confidence intervals, we first calculate the variance of the reliability, <math>R\,\!</math>, using:		If both the stress and strength distributions are estimated from data sets, then there are uncertainties associated with the estimated distribution parameters. These uncertainties will cause some degree of variation of the probability calculated from the stress-strength analysis. Therefore, we can use these uncertainties to estimate the confidence intervals on the calculated probability. To get the confidence intervals, we first calculate the variance of the reliability, <math>R\,\!</math>, using:

	::<math>Var\left[ R \right]=\int_{0}^{\infty }{Var\left[ {{f}_{1}}(x) \right]}{{\left[ {{R}_{2}}(x) \right]}^{2}}dx+{{\int_{0}^{\infty }{\left[ {{f}_{1}}(x) \right]}}^{2}}Var\left[ {{R}_{2}}(x) \right]dx</math>		::<math>Var\left[ R \right]=\int_{0}^{\infty }{Var\left[ {{f}_{1}}(x) \right]}{{\left[ {{R}_{2}}(x) \right]}^{2}}dx+{{\int_{0}^{\infty }{\left[ {{f}_{1}}(x) \right]}}^{2}}Var\left[ {{R}_{2}}(x) \right]dx\,\!</math>

	Variance of <math>{{f}_{1}}(x)\,\!</math> and <math>{{R}_{2}}(x)\,\!</math> can be estimated from the Fisher Information Matrix. For details, please see [[Confidence Bounds]].		Variance of <math>{{f}_{1}}(x)\,\!</math> and <math>{{R}_{2}}(x)\,\!</math> can be estimated from the Fisher Information Matrix. For details, please see [[Confidence Bounds]].
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	Once the variance of the expected reliability is obtained, the two-sided confidence intervals can be calculated using:		Once the variance of the expected reliability is obtained, the two-sided confidence intervals can be calculated using:

	::<math>[\frac{R}{R+(1-R)w},\frac{R}{R+(1-R)/w}]</math>		::<math>[\frac{R}{R+(1-R)w},\frac{R}{R+(1-R)/w}]\,\!</math>

	where:		where:
	:: CL is the confidence level		:: CL is the confidence level
	:: <math>\alpha</math> is 1-CL		:: <math>\alpha\,\!</math> is 1-CL
	:: <math>w=\exp \{{{z}_{1-\alpha /2}}\sqrt{Var(R)}/[R(1-R)]\}</math>		:: <math>w=\exp \{{{z}_{1-\alpha /2}}\sqrt{Var(R)}/[R(1-R)]\}\,\!</math>
	:: <math>{{Z}_{1-\alpha /2}}</math> is the <math>1-\alpha/2</math> percentile of a standard normal distribution.		:: <math>{{Z}_{1-\alpha /2}}\,\!</math> is the <math>1-\alpha/2\,\!</math> percentile of a standard normal distribution.

	If the upper bound (U) and lower bound (L) are not infinite and 0, respectively, then the calculated variance of <math> R \,\!</math> is adjusted by <math>{{\left[ {1}/{\left( {{F}_{1}}(U)-{{F}_{1}}(L) \right)}\; \right]}^{2}}\,\!</math>.		If the upper bound (U) and lower bound (L) are not infinite and 0, respectively, then the calculated variance of <math> R \,\!</math> is adjusted by <math>{{\left[ {1}/{\left( {{F}_{1}}(U)-{{F}_{1}}(L) \right)}\; \right]}^{2}}\,\!</math>.
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	Assume the distributions for stress and strength are known. From the stress-strength equation		Assume the distributions for stress and strength are known. From the stress-strength equation

	::<math>R=P[Stress\le Strength]=\int_{0}^{\infty }{{{f}_{Stress}}(x)\cdot {{R}_{Strength}}(x)}dx</math>		::<math>R=P[Stress\le Strength]=\int_{0}^{\infty }{{{f}_{Stress}}(x)\cdot {{R}_{Strength}}(x)}dx\,\!</math>

	we know that the calculated reliability is the '''expected''' value of the probability that a strength value is larger than a stress value. Since both strength and stress are random variables from their distributions, the '''reliability''' is also a random variable. This can be explained using the following example. Let's first assume that stress is a fixed value of 567. The reliability then is:		we know that the calculated reliability is the '''expected''' value of the probability that a strength value is larger than a stress value. Since both strength and stress are random variables from their distributions, the '''reliability''' is also a random variable. This can be explained using the following example. Let's first assume that stress is a fixed value of 567. The reliability then is:

	::<math>R(567)=\Pr (Strength>567)={{R}_{2}}(567)</math>		::<math>R(567)=\Pr (Strength>567)={{R}_{2}}(567)\,\!</math>

	This is the reliability when the stress value is 567 and when the strength distribution is given. If stress is not a fixed value (i.e., it follows a distribution instead), then it can take values other than 567. For instance, if stress takes a value of 700, then we get another reliability value of <math>{R}(700)\,\!</math>. Since stress is a random variable, for any stress value <math>{x}_{i}\,\!</math>, there is a reliability value of		This is the reliability when the stress value is 567 and when the strength distribution is given. If stress is not a fixed value (i.e., it follows a distribution instead), then it can take values other than 567. For instance, if stress takes a value of 700, then we get another reliability value of <math>{R}(700)\,\!</math>. Since stress is a random variable, for any stress value <math>{x}_{i}\,\!</math>, there is a reliability value of
	<math>R({{x}_{i}})\,\!</math> calculated from the strength distribution. We will end up with many <math>R({{x}_{i}})\,\!</math>s or <math>R_{2}({{x}_{i}})\,\!</math>s. From these <math>R({{x}_{i}})\,\!</math>s, we can get the mean and variance of the reliability. In fact, its mean is the result from:		<math>R({{x}_{i}})\,\!</math> calculated from the strength distribution. We will end up with many <math>R({{x}_{i}})\,\!</math>s or <math>R_{2}({{x}_{i}})\,\!</math>s. From these <math>R({{x}_{i}})\,\!</math>s, we can get the mean and variance of the reliability. In fact, its mean is the result from:

	::<math>R=P[Stress\le Strength]=\int_{0}^{\infty }{{{f}_{Stress}}(x)\cdot {{R}_{Strength}}(x)}dx</math>		::<math>R=P[Stress\le Strength]=\int_{0}^{\infty }{{{f}_{Stress}}(x)\cdot {{R}_{Strength}}(x)}dx\,\!</math>

	and its variance is:		and its variance is:
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	& =\int_{0}^{\infty }{{{f}_{1}}(x){{\left[ {{R}_{2}}(x) \right]}^{2}}dx}-{{\left( E\left[ {{R}_{2}}\left( {{X}_{1}} \right) \right] \right)}^{2}} \\		& =\int_{0}^{\infty }{{{f}_{1}}(x){{\left[ {{R}_{2}}(x) \right]}^{2}}dx}-{{\left( E\left[ {{R}_{2}}\left( {{X}_{1}} \right) \right] \right)}^{2}} \\
	& =\int_{0}^{\infty }{{{f}_{1}}(x){{\left[ {{R}_{2}}(x) \right]}^{2}}dx}-{{\left( R \right)}^{2}} \\		& =\int_{0}^{\infty }{{{f}_{1}}(x){{\left[ {{R}_{2}}(x) \right]}^{2}}dx}-{{\left( R \right)}^{2}} \\
	\end{align}</math>		\end{align}\,\!</math>

	where ''R'' is the expected value of the reliability.		where ''R'' is the expected value of the reliability.
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	Once the variance of the expected reliability is obtained, the two-sided confidence intervals can be calculated using:		Once the variance of the expected reliability is obtained, the two-sided confidence intervals can be calculated using:

	::<math>[\frac{R}{R+(1-R)w},\frac{R}{R+(1-R)/w}]</math>		::<math>[\frac{R}{R+(1-R)w},\frac{R}{R+(1-R)/w}]\,\!</math>

	where:		where:
	:: CL is the confidence level		:: CL is the confidence level
	:: <math>\alpha</math> is 1-CL		:: <math>\alpha\,\!</math> is 1-CL
	:: <math>w=\exp \{{{z}_{1-\alpha /2}}\sqrt{Var(R)}/[R(1-R)]\}</math>		:: <math>w=\exp \{{{z}_{1-\alpha /2}}\sqrt{Var(R)}/[R(1-R)]\}\,\!</math>
	:: <math>{{Z}_{1-\alpha /2}}</math> is the <math>1-\alpha/2</math> percentile of a standard normal distribution		:: <math>{{Z}_{1-\alpha /2}}\,\!</math> is the <math>1-\alpha/2\,\!</math> percentile of a standard normal distribution


	If the upper bound (U) and lower bound (L) are not infinite and 0, the above calculated variance of ''R'' is adjusted by <math>{{\left[ {1}/{\left( {{F}_{1}}(U)-{{F}_{1}}(L) \right)}\; \right]}^{2}}</math>.		If the upper bound (U) and lower bound (L) are not infinite and 0, the above calculated variance of ''R'' is adjusted by <math>{{\left[ {1}/{\left( {{F}_{1}}(U)-{{F}_{1}}(L) \right)}\; \right]}^{2}}\,\!</math>.

	===Example 1===		===Example 1===
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	As we know, the expected reliability is called from the following stress-strength calculation:		As we know, the expected reliability is called from the following stress-strength calculation:

	::<math>R=P[Stress\le Strength]=\int_{0}^{\infty }{{{f}_{Stress}}(x)\cdot {{R}_{Strength}}(x)}dx</math>		::<math>R=P[Stress\le Strength]=\int_{0}^{\infty }{{{f}_{Stress}}(x)\cdot {{R}_{Strength}}(x)}dx\,\!</math>

	The stress distribution is usually estimated from customer usage data, such as the mileage per year of a passenger car or the load distribution for a beam. The strength distribution, on the other hand, is affected by the material used in the component, the geometric dimensions and the manufacturing process.		The stress distribution is usually estimated from customer usage data, such as the mileage per year of a passenger car or the load distribution for a beam. The strength distribution, on the other hand, is affected by the material used in the component, the geometric dimensions and the manufacturing process.