Instance Models

This section transforms the problem defined in the problem description into one which is expressed in mathematical terms. It uses concrete symbols defined in the data definitions to replace the abstract symbols in the models identified in theoretical models and general definitions.

The goal GS:Predict-Glass-Withstands-Explosion is met by IM:isSafePb, IM:isSafeLR.

Risk of failure

Refname	IM:riskFun
Label	Risk of failure
Input	$E$ , $\mathit{LDF}$ , $J$ , $k$ , $m$ , $h$ , $a$ , $b$
Output	$B$
Input Constraints	$a\gt{}0$ $0\lt{}b\leq{}a$
Output Constraints
Equation	$B=\frac{k}{\left(a\,b\right)^{m-1}}\,\left(E\,h^{2}\right)^{m}\,\mathit{LDF}\,e^{J}$
Description	$B$ is the risk of failure (Unitless) $k$ is the surface flaw parameter ( $\frac{\text{m}^{12}}{\text{N}^{7}}$ ) $a$ is the plate length (long dimension) ( ${\text{m}}$ ) $b$ is the plate width (short dimension) ( ${\text{m}}$ ) $m$ is the surface flaw parameter ( $\frac{\text{m}^{12}}{\text{N}^{7}}$ ) $E$ is the modulus of elasticity of glass ( ${\text{Pa}}$ ) $h$ is the minimum thickness ( ${\text{m}}$ ) $\mathit{LDF}$ is the load duration factor (Unitless) $J$ is the stress distribution factor (Function) (Unitless)
Notes	$a$ and $b$ are the dimensions of the plate, where ( $a\geq{}b$ ). $h$ is defined in DD:minThick and is based on the nominal thicknesses. $\mathit{LDF}$ is defined in DD:loadDurFactor. $J$ is defined in IM:stressDistFac.
Source	astm2009, beasonEtAl1998 (Eqs. 4-5), and campidelli (Eq. 14)
RefBy	IM:probOfBreak

Stress distribution factor (Function)

Refname	IM:stressDistFac
Label	Stress distribution factor (Function)
Input	$\mathit{AR}$ , $\hat{q}$
Output	$J$
Input Constraints	$\mathit{AR}\geq{}1$
Output Constraints	${J_{\text{min}}}\leq{}J\leq{}{J_{\text{max}}}$
Equation	$J=\mathit{interpZ}\left(\text{$``$SDF.txt’’},\mathit{AR},\hat{q}\right)$
Description	$J$ is the stress distribution factor (Function) (Unitless) $\mathit{interpZ}$ is the interpZ (Unitless) $\mathit{AR}$ is the aspect ratio (Unitless) $\hat{q}$ is the dimensionless load (Unitless)
Notes	$J$ is obtained by interpolating from data shown in Fig:dimlessloadVSaspect. $\mathit{AR}$ is defined in DD:aspectRatio. $\hat{q}$ is defined in IM:dimlessLoad.
Source	astm2009
RefBy	IM:riskFun

Non-factored load

Refname	IM:nFL
Label	Non-factored load
Input	${\hat{q}_{\text{tol}}}$ , $E$ , $h$ , $a$ , $b$
Output	$\mathit{NFL}$
Input Constraints	$a\gt{}0$ $0\lt{}b\leq{}a$
Output Constraints
Equation	$\mathit{NFL}=\frac{{\hat{q}_{\text{tol}}}\,E\,h^{4}}{\left(a\,b\right)^{2}}$
Description	$\mathit{NFL}$ is the non-factored load ( ${\text{Pa}}$ ) ${\hat{q}_{\text{tol}}}$ is the tolerable load (Unitless) $E$ is the modulus of elasticity of glass ( ${\text{Pa}}$ ) $h$ is the minimum thickness ( ${\text{m}}$ ) $a$ is the plate length (long dimension) ( ${\text{m}}$ ) $b$ is the plate width (short dimension) ( ${\text{m}}$ )
Notes	${\hat{q}_{\text{tol}}}$ is defined in IM:tolLoad. $E$ comes from A:standardValues. $h$ is defined in DD:minThick and is based on the nominal thicknesses. $a$ and $b$ are the dimensions of the plate, where ( $a\geq{}b$ ).
Source	astm2009
RefBy	IM:calofCapacity

Dimensionless load

Refname	IM:dimlessLoad
Label	Dimensionless load
Input	$q$ , $E$ , $h$ , $\mathit{GTF}$ , $a$ , $b$
Output	$\hat{q}$
Input Constraints	$a\gt{}0$ $0\lt{}b\leq{}a$
Output Constraints
Equation	$\hat{q}=\frac{q\,\left(a\,b\right)^{2}}{E\,h^{4}\,\mathit{GTF}}$
Description	$\hat{q}$ is the dimensionless load (Unitless) $q$ is the applied load (demand) ( ${\text{Pa}}$ ) $a$ is the plate length (long dimension) ( ${\text{m}}$ ) $b$ is the plate width (short dimension) ( ${\text{m}}$ ) $E$ is the modulus of elasticity of glass ( ${\text{Pa}}$ ) $h$ is the minimum thickness ( ${\text{m}}$ ) $\mathit{GTF}$ is the glass type factor (Unitless)
Notes	$q$ is the 3 second duration equivalent pressure, as given in DD:calofDemand. $a$ and $b$ are the dimensions of the plate, where ( $a\geq{}b$ ). $E$ comes from A:standardValues. $h$ is defined in DD:minThick and is based on the nominal thicknesses. $\mathit{GTF}$ is defined in DD:gTF.
Source	astm2009 and campidelli (Eq. 7)
RefBy	IM:stressDistFac

Tolerable load

Refname	IM:tolLoad
Label	Tolerable load
Input	$\mathit{AR}$ , ${J_{\text{tol}}}$
Output	${\hat{q}_{\text{tol}}}$
Input Constraints	$\mathit{AR}\geq{}1$
Output Constraints
Equation	${\hat{q}_{\text{tol}}}=\mathit{interpY}\left(\text{$``$SDF.txt’’},\mathit{AR},{J_{\text{tol}}}\right)$
Description	${\hat{q}_{\text{tol}}}$ is the tolerable load (Unitless) $\mathit{interpY}$ is the interpY (Unitless) $\mathit{AR}$ is the aspect ratio (Unitless) ${J_{\text{tol}}}$ is the tolerable stress distribution factor (Unitless)
Notes	${\hat{q}_{\text{tol}}}$ is obtained by interpolating from data shown in Fig:dimlessloadVSaspect. $\mathit{AR}$ is defined in DD:aspectRatio. ${J_{\text{tol}}}$ is defined in IM:sdfTol.
Source	astm2009
RefBy	IM:nFL

Tolerable stress distribution factor

Refname	IM:sdfTol
Label	Tolerable stress distribution factor
Input	$\mathit{LDF}$ , ${P_{\text{b}\text{tol}}}$ , $E$ , $a$ , $b$ , $m$ , $k$ , $h$
Output	${J_{\text{tol}}}$
Input Constraints	$0\leq{}{P_{\text{b}\text{tol}}}\leq{}1$ $a\gt{}0$ $0\lt{}b\leq{}a$
Output Constraints
Equation	${J_{\text{tol}}}=\ln\left(\ln\left(\frac{1}{1-{P_{\text{b}\text{tol}}}}\right)\,\frac{\left(a\,b\right)^{m-1}}{k\,\left(E\,h^{2}\right)^{m}\,\mathit{LDF}}\right)$
Description	${J_{\text{tol}}}$ is the tolerable stress distribution factor (Unitless) ${P_{\text{b}\text{tol}}}$ is the tolerable probability of breakage (Unitless) $a$ is the plate length (long dimension) ( ${\text{m}}$ ) $b$ is the plate width (short dimension) ( ${\text{m}}$ ) $m$ is the surface flaw parameter ( $\frac{\text{m}^{12}}{\text{N}^{7}}$ ) $k$ is the surface flaw parameter ( $\frac{\text{m}^{12}}{\text{N}^{7}}$ ) $E$ is the modulus of elasticity of glass ( ${\text{Pa}}$ ) $h$ is the minimum thickness ( ${\text{m}}$ ) $\mathit{LDF}$ is the load duration factor (Unitless)
Notes	${P_{\text{b}\text{tol}}}$ is entered by the user. $a$ and $b$ are the dimensions of the plate, where ( $a\geq{}b$ ). $m$ , $k$ , and $E$ come from A:standardValues. $h$ is defined in DD:minThick and is based on the nominal thicknesses. $\mathit{LDF}$ is defined in DD:loadDurFactor.
Source	astm2009
RefBy	IM:tolLoad

Probability of breakage

Refname	IM:probOfBreak
Label	Probability of breakage
Input	$B$
Output	${P_{\text{b}}}$
Input Constraints
Output Constraints	$0\leq{}{P_{\text{b}}}\leq{}1$
Equation	${P_{\text{b}}}=1-e^{-B}$
Description	${P_{\text{b}}}$ is the probability of breakage (Unitless) $B$ is the risk of failure (Unitless)
Notes	$B$ is defined in IM:riskFun.
Source	astm2009 and beasonEtAl1998
RefBy	IM:isSafePb

Load resistance

Refname	IM:calofCapacity
Label	Load resistance
Input	$\mathit{NFL}$ , $\mathit{GTF}$ , $\mathit{LSF}$
Output	$\mathit{LR}$
Input Constraints
Output Constraints
Equation	$\mathit{LR}=\mathit{NFL}\,\mathit{GTF}\,\mathit{LSF}$
Description	$\mathit{LR}$ is the load resistance ( ${\text{Pa}}$ ) $\mathit{NFL}$ is the non-factored load ( ${\text{Pa}}$ ) $\mathit{GTF}$ is the glass type factor (Unitless) $\mathit{LSF}$ is the load share factor (Unitless)
Notes	$\mathit{LR}$ is also called capacity. $\mathit{NFL}$ is defined in IM:nFL. $\mathit{GTF}$ is defined in DD:gTF.
Source	astm2009
RefBy	IM:isSafeLR

Safety Req-Pb

Refname	IM:isSafePb
Label	Safety Req-Pb
Input	${P_{\text{b}}}$ , ${P_{\text{b}\text{tol}}}$
Output	$\mathit{isSafePb}$
Input Constraints	$0\leq{}{P_{\text{b}}}\leq{}1$ $0\leq{}{P_{\text{b}\text{tol}}}\leq{}1$
Output Constraints
Equation	$\mathit{isSafePb}={P_{\text{b}}}\lt{}{P_{\text{b}\text{tol}}}$
Description	$\mathit{isSafePb}$ is the probability of glass breakage safety requirement (Unitless) ${P_{\text{b}}}$ is the probability of breakage (Unitless) ${P_{\text{b}\text{tol}}}$ is the tolerable probability of breakage (Unitless)
Notes	If $\mathit{isSafePb}$ , the glass is considered safe. $\mathit{isSafePb}$ and $\mathit{isSafeLR}$ (from IM:isSafeLR) are either both True or both False. ${P_{\text{b}}}$ is defined in IM:probOfBreak. ${P_{\text{b}\text{tol}}}$ is entered by the user.
Source	astm2009
RefBy	IM:isSafeLR and FR:Check-Glass-Safety

Safety Req-LR

Refname	IM:isSafeLR
Label	Safety Req-LR
Input	$\mathit{LR}$ , $q$
Output	$\mathit{isSafeLR}$
Input Constraints	$\mathit{LR}\gt{}0$ $q\gt{}0$
Output Constraints
Equation	$\mathit{isSafeLR}=\mathit{LR}\gt{}q$
Description	$\mathit{isSafeLR}$ is the 3 second load equivalent resistance safety requirement (Unitless) $\mathit{LR}$ is the load resistance ( ${\text{Pa}}$ ) $q$ is the applied load (demand) ( ${\text{Pa}}$ )
Notes	If $\mathit{isSafeLR}$ , the glass is considered safe. $\mathit{isSafePb}$ (from IM:isSafePb) and $\mathit{isSafeLR}$ are either both True or both False. $\mathit{LR}$ is defined in IM:calofCapacity and is also called capacity. $q$ is the 3 second duration equivalent pressure, as given in DD:calofDemand.
Source	astm2009
RefBy	IM:isSafePb and FR:Check-Glass-Safety

Software Requirements Specification for GlassBR