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 .
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
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 Source astm2009 RefBy IM:riskFun
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
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
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 Source astm2009 RefBy IM:nFL
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
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 Source astm2009 and beasonEtAl1998 RefBy IM:isSafePb
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
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
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