diff --git a/code/drasil-printers/Language/Drasil/TeX/Print.hs b/code/drasil-printers/Language/Drasil/TeX/Print.hs index 8ce0cf1f38..555544417b 100644 --- a/code/drasil-printers/Language/Drasil/TeX/Print.hs +++ b/code/drasil-printers/Language/Drasil/TeX/Print.hs @@ -108,7 +108,7 @@ pExpr (Mtx a) = mkEnv "bmatrix" (pMatrix a) pExpr (Row [x]) = br $ pExpr x -- FIXME: Hack needed for symbols with multiple subscripts, etc. pExpr (Row l) = foldl1 (<>) (map pExpr l) pExpr (Ident s) = pure . text $ s -pExpr (Label s) = pure . text $ s -- command "text" s +pExpr (Label s) = command "text" s pExpr (Spec s) = pure . text $ unPL $ L.special s --pExpr (Gr g) = unPL $ greek g pExpr (Sub e) = pure unders <> br (pExpr e) diff --git a/code/stable/gamephys/SRS/Chipmunk_SRS.tex b/code/stable/gamephys/SRS/Chipmunk_SRS.tex index 775e3d9f16..e7b1a0167b 100644 --- a/code/stable/gamephys/SRS/Chipmunk_SRS.tex +++ b/code/stable/gamephys/SRS/Chipmunk_SRS.tex @@ -66,7 +66,7 @@ \subsection{Table of Symbols} \endhead $\mathbf{a}$ & Acceleration & $\frac{\text{m}}{\text{s}^{2}}$ \\ -$\mathbf{a}(t)$ & Linear Acceleration & $\frac{\text{m}}{\text{s}^{2}}$ +$\mathbf{a}\text{(}t\text{)}$ & Linear Acceleration & $\frac{\text{m}}{\text{s}^{2}}$ \\ ${\mathbf{a}_{i}}$ & The I-Th Body's Acceleration & $\frac{\text{m}}{\text{s}^{2}}$ \\ @@ -74,9 +74,9 @@ \subsection{Table of Symbols} \\ $\mathbf{F}$ & Force & N \\ -${\mathbf{F}_{1}}$ & Force exerted by the first body (on another body) & N +${\mathbf{F}_{\text{1}}}$ & Force exerted by the first body (on another body) & N \\ -${\mathbf{F}_{2}}$ & Force exerted by the second body (on another body) & N +${\mathbf{F}_{\text{2}}}$ & Force exerted by the second body (on another body) & N \\ ${\mathbf{F}_{i}}$ & Force Applied to the I-Th Body at Time T & N \\ @@ -88,9 +88,9 @@ \subsection{Table of Symbols} \\ $\mathbf{I}$ & Moment of inertia & kg$\text{m}^{2}$ \\ -${\mathbf{I}_{A}}$ & Moment of Inertia Of Rigid Body A & kg$\text{m}^{2}$ +${\mathbf{I}_{\text{A}}}$ & Moment of Inertia Of Rigid Body A & kg$\text{m}^{2}$ \\ -${\mathbf{I}_{B}}$ & Moment of Inertia Of Rigid Body B & kg$\text{m}^{2}$ +${\mathbf{I}_{\text{B}}}$ & Moment of Inertia Of Rigid Body B & kg$\text{m}^{2}$ \\ $\mathbf{J}$ & Impulse (vector) & Ns \\ @@ -104,9 +104,9 @@ \subsection{Table of Symbols} \\ $m$ & Mass & kg \\ -${m_{1}}$ & Mass of the first body & kg +${m_{\text{1}}}$ & Mass of the first body & kg \\ -${m_{2}}$ & Mass of the second body & kg +${m_{\text{2}}}$ & Mass of the second body & kg \\ ${m_{A}}$ & Mass Of Rigid Body A & kg \\ @@ -120,7 +120,7 @@ \subsection{Table of Symbols} \\ $\mathbf{p}$ & Position & m \\ -${\mathbf{p}_{CM}}$ & Center of Mass & m +${\mathbf{p}_{\text{CM}}}$ & Center of Mass & m \\ ${\mathbf{p}_{j}}$ & Position Vector of the J-Th Particle & m \\ @@ -128,7 +128,7 @@ \subsection{Table of Symbols} \\ $\mathbf{r}$ & Displacement & m \\ -$\mathbf{r}(t)$ & Linear Displacement & m +$\mathbf{r}\text{(}t\text{)}$ & Linear Displacement & m \\ ${\mathbf{r}_{OB}}$ & Displacement vector between the origin and point B & m \\ @@ -142,15 +142,15 @@ \subsection{Table of Symbols} \\ $Δ\mathbf{v}$ & Change in velocity & $\frac{\text{m}}{\text{s}}$ \\ -$\mathbf{v}(t)$ & Linear Velocity & $\frac{\text{m}}{\text{s}}$ +$\mathbf{v}\text{(}t\text{)}$ & Linear Velocity & $\frac{\text{m}}{\text{s}}$ \\ ${\mathbf{v}^{AP}}$ & Velocity Of the Point of Collision P in Body A & $\frac{\text{m}}{\text{s}}$ \\ ${\mathbf{v}^{BP}}$ & Velocity Of the Point of Collision P in Body B & $\frac{\text{m}}{\text{s}}$ \\ -${\mathbf{v}_{1}}$ & Velocity Of the First Body & $\frac{\text{m}}{\text{s}}$ +${\mathbf{v}_{\text{1}}}$ & Velocity Of the First Body & $\frac{\text{m}}{\text{s}}$ \\ -${\mathbf{v}_{2}}$ & Velocity Of the Second Body & $\frac{\text{m}}{\text{s}}$ +${\mathbf{v}_{\text{2}}}$ & Velocity Of the Second Body & $\frac{\text{m}}{\text{s}}$ \\ ${\mathbf{v}_{A}}$ & Velocity At Point A & $\frac{\text{m}}{\text{s}}$ \\ @@ -164,15 +164,15 @@ \subsection{Table of Symbols} \\ ${\mathbf{v}_{O}}$ & Velocity At Point Origin & $\frac{\text{m}}{\text{s}}$ \\ -$||\mathbf{n}||$ & Length of the Normal Vector & m +$\text{||}\mathbf{n}\text{||}$ & Length of the Normal Vector & m \\ -$||\mathbf{r}||$ & Euclidean norm of the displacement & m +$\text{||}\mathbf{r}\text{||}$ & Euclidean norm of the displacement & m \\ -$||{\mathbf{r}_{AP}}*\mathbf{n}||$ & Length of the Perpendicular Vector To the Contact Displacement Vector of Rigid Body A & m +$\text{||}{\mathbf{r}_{\text{A}P}}\text{*}\mathbf{n}\text{||}$ & Length of the Perpendicular Vector To the Contact Displacement Vector of Rigid Body A & m \\ -$||{\mathbf{r}_{BP}}*\mathbf{n}||$ & Length of the Perpendicular Vector To the Contact Displacement Vector of Rigid Body B & m +$\text{||}{\mathbf{r}_{\text{B}P}}\text{*}\mathbf{n}\text{||}$ & Length of the Perpendicular Vector To the Contact Displacement Vector of Rigid Body B & m \\ -${||\mathbf{r}||^{2}}$ & Squared distance & $\text{m}^{2}$ +${\text{||}\mathbf{r}\text{||}^{\text{2}}}$ & Squared distance & $\text{m}^{2}$ \\ $α$ & Angular Acceleration & $\frac{\text{rad}}{\text{s}^{2}}$ \\ @@ -366,15 +366,15 @@ \subsubsection{Theoretical Models} Label & Newton's third law of motion \\ \midrule \\ Equation & \begin{displaymath} - {\mathbf{F}_{1}}=-{\mathbf{F}_{2}} + {\mathbf{F}_{\text{1}}}=-{\mathbf{F}_{\text{2}}} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} - \item{${\mathbf{F}_{1}}$ is the force exerted by the first body (on another body) (N)} - \item{${\mathbf{F}_{2}}$ is the force exerted by the second body (on another body) (N)} + \item{${\mathbf{F}_{\text{1}}}$ is the force exerted by the first body (on another body) (N)} + \item{${\mathbf{F}_{\text{2}}}$ is the force exerted by the second body (on another body) (N)} \end{symbDescription} \\ \midrule \\ -Notes & Every action has an equal and opposite reaction. In other words, the force ${\mathbf{F}_{1}}$ (N) exerted on the second rigid body by the first is equal in magnitude and in the opposite direction to the force ${\mathbf{F}_{2}}$ (N) exerted on the first rigid body by the second. +Notes & Every action has an equal and opposite reaction. In other words, the force ${\mathbf{F}_{\text{1}}}$ (N) exerted on the second rigid body by the first is equal in magnitude and in the opposite direction to the force ${\mathbf{F}_{\text{2}}}$ (N) exerted on the first rigid body by the second. \\ \midrule \\ Source & -- \\ \midrule \\ @@ -393,20 +393,20 @@ \subsubsection{Theoretical Models} Label & Newton's law of universal gravitation \\ \midrule \\ Equation & \begin{displaymath} - \mathbf{F}=G \frac{{m_{1}} {m_{2}}}{||\mathbf{r}||^{2}} \mathbf{\hat{r}}=G \frac{{m_{1}} {m_{2}}}{||\mathbf{r}||^{2}} \frac{\mathbf{r}}{||\mathbf{r}||} + \mathbf{F}=G \frac{{m_{\text{1}}} {m_{\text{2}}}}{\text{||}\mathbf{r}\text{||}^{2}} \mathbf{\hat{r}}=G \frac{{m_{\text{1}}} {m_{\text{2}}}}{\text{||}\mathbf{r}\text{||}^{2}} \frac{\mathbf{r}}{\text{||}\mathbf{r}\text{||}} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} \item{$\mathbf{F}$ is the force (N)} \item{$G$ is the gravitational constant ($\frac{\text{m}^{3}}{(\text{kg}\text{s}^{2})}$)} - \item{${m_{1}}$ is the mass of the first body (kg)} - \item{${m_{2}}$ is the mass of the second body (kg)} - \item{$||\mathbf{r}||$ is the Euclidean norm of the displacement (m)} + \item{${m_{\text{1}}}$ is the mass of the first body (kg)} + \item{${m_{\text{2}}}$ is the mass of the second body (kg)} + \item{$\text{||}\mathbf{r}\text{||}$ is the Euclidean norm of the displacement (m)} \item{$\mathbf{\hat{r}}$ is the displacement unit vector (m)} \item{$\mathbf{r}$ is the displacement (m)} \end{symbDescription} \\ \midrule \\ -Notes & Two rigid bodies in the universe attract each other with a force $\mathbf{F}$ (N) that is directly proportional to the product of their masses, ${m_{1}}$ and ${m_{2}}$ (kg), and inversely proportional to the squared distance ${||\mathbf{r}||^{2}}$ ($\text{m}^{2}$) between them. The vector $\mathbf{r}$ (m) is the displacement between the centres of the rigid bodies and $||\mathbf{r}||$ (m) represents the Euclidean norm of the displacement, or absolute distance between the two. $\mathbf{\hat{r}}$ denotes the displacement unit vector, equivalent to the displacement divided by the Euclidean norm of the displacement, as shown above. Finally, $G$ is the gravitational constant (6.673 * 10E-11) ($\frac{\text{m}^{3}}{(\text{kg}\text{s}^{2})}$). +Notes & Two rigid bodies in the universe attract each other with a force $\mathbf{F}$ (N) that is directly proportional to the product of their masses, ${m_{\text{1}}}$ and ${m_{\text{2}}}$ (kg), and inversely proportional to the squared distance ${\text{||}\mathbf{r}\text{||}^{\text{2}}}$ ($\text{m}^{2}$) between them. The vector $\mathbf{r}$ (m) is the displacement between the centres of the rigid bodies and $\text{||}\mathbf{r}\text{||}$ (m) represents the Euclidean norm of the displacement, or absolute distance between the two. $\mathbf{\hat{r}}$ denotes the displacement unit vector, equivalent to the displacement divided by the Euclidean norm of the displacement, as shown above. Finally, $G$ is the gravitational constant (6.673 * 10E-11) ($\frac{\text{m}^{3}}{(\text{kg}\text{s}^{2})}$). \\ \midrule \\ Source & -- \\ \midrule \\ @@ -487,16 +487,16 @@ \subsubsection{Data Definitions} \\ \midrule \\ Label & Center of Mass \\ \midrule \\ -Symbol & ${\mathbf{p}_{CM}}$ +Symbol & ${\mathbf{p}_{\text{CM}}}$ \\ \midrule \\ Units & m \\ \midrule \\ Equation & \begin{displaymath} - {\mathbf{p}_{CM}}=\frac{\displaystyle\sum{{m_{j}} {\mathbf{p}_{j}}}}{M} + {\mathbf{p}_{\text{CM}}}=\frac{\displaystyle\sum{{m_{j}} {\mathbf{p}_{j}}}}{M} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} - \item{${\mathbf{p}_{CM}}$ is the Center of Mass (m)} + \item{${\mathbf{p}_{\text{CM}}}$ is the Center of Mass (m)} \item{${m_{j}}$ is the mass of the j-th particle (kg)} \item{${\mathbf{p}_{j}}$ is the position vector of the j-th particle (m)} \item{$M$ is the total mass of the rigid body (kg)} @@ -521,16 +521,16 @@ \subsubsection{Data Definitions} \\ \midrule \\ Label & Linear Displacement \\ \midrule \\ -Symbol & $\mathbf{r}(t)$ +Symbol & $\mathbf{r}\text{(}t\text{)}$ \\ \midrule \\ Units & m \\ \midrule \\ Equation & \begin{displaymath} - \mathbf{r}(t)=\frac{\,d\mathbf{p}\left(t\right)}{\,dt} + \mathbf{r}\text{(}t\text{)}=\frac{\,d\mathbf{p}\left(t\right)}{\,dt} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} - \item{$\mathbf{r}(t)$ is the linear displacement (m)} + \item{$\mathbf{r}\text{(}t\text{)}$ is the linear displacement (m)} \item{$t$ is the time (s)} \item{$\mathbf{p}$ is the position (m)} \end{symbDescription} @@ -555,16 +555,16 @@ \subsubsection{Data Definitions} \\ \midrule \\ Label & Linear Velocity \\ \midrule \\ -Symbol & $\mathbf{v}(t)$ +Symbol & $\mathbf{v}\text{(}t\text{)}$ \\ \midrule \\ Units & $\frac{\text{m}}{\text{s}}$ \\ \midrule \\ Equation & \begin{displaymath} - \mathbf{v}(t)=\frac{\,d\mathbf{r}\left(t\right)}{\,dt} + \mathbf{v}\text{(}t\text{)}=\frac{\,d\mathbf{r}\left(t\right)}{\,dt} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} - \item{$\mathbf{v}(t)$ is the linear velocity ($\frac{\text{m}}{\text{s}}$)} + \item{$\mathbf{v}\text{(}t\text{)}$ is the linear velocity ($\frac{\text{m}}{\text{s}}$)} \item{$t$ is the time (s)} \item{$\mathbf{r}$ is the displacement (m)} \end{symbDescription} @@ -589,16 +589,16 @@ \subsubsection{Data Definitions} \\ \midrule \\ Label & Linear Acceleration \\ \midrule \\ -Symbol & $\mathbf{a}(t)$ +Symbol & $\mathbf{a}\text{(}t\text{)}$ \\ \midrule \\ Units & $\frac{\text{m}}{\text{s}^{2}}$ \\ \midrule \\ Equation & \begin{displaymath} - \mathbf{a}(t)=\frac{\,d\mathbf{v}\left(t\right)}{\,dt} + \mathbf{a}\text{(}t\text{)}=\frac{\,d\mathbf{v}\left(t\right)}{\,dt} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} - \item{$\mathbf{a}(t)$ is the linear acceleration ($\frac{\text{m}}{\text{s}^{2}}$)} + \item{$\mathbf{a}\text{(}t\text{)}$ is the linear acceleration ($\frac{\text{m}}{\text{s}^{2}}$)} \item{$t$ is the time (s)} \item{$\mathbf{v}$ is the velocity ($\frac{\text{m}}{\text{s}}$)} \end{symbDescription} @@ -730,7 +730,7 @@ \subsubsection{Data Definitions} Units & Ns \\ \midrule \\ Equation & \begin{displaymath} - j=\frac{-\left(1+{C_{R}}\right) {{\mathbf{v}_{i}}^{AB}}\cdot{}\mathbf{n}}{\left(\frac{1}{{m_{A}}}+\frac{1}{{m_{B}}}\right) ||\mathbf{n}||^{2}+\frac{||{\mathbf{r}_{AP}}*\mathbf{n}||^{2}}{{\mathbf{I}_{A}}}+\frac{||{\mathbf{r}_{BP}}*\mathbf{n}||^{2}}{{\mathbf{I}_{B}}}} + j=\frac{-\left(1+{C_{R}}\right) {{\mathbf{v}_{i}}^{AB}}\cdot{}\mathbf{n}}{\left(\frac{1}{{m_{A}}}+\frac{1}{{m_{B}}}\right) \text{||}\mathbf{n}\text{||}^{2}+\frac{\text{||}{\mathbf{r}_{\text{A}P}}\text{*}\mathbf{n}\text{||}^{2}}{{\mathbf{I}_{\text{A}}}}+\frac{\text{||}{\mathbf{r}_{\text{B}P}}\text{*}\mathbf{n}\text{||}^{2}}{{\mathbf{I}_{\text{B}}}}} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} @@ -740,11 +740,11 @@ \subsubsection{Data Definitions} \item{$\mathbf{n}$ is the collision normal vector (m)} \item{${m_{A}}$ is the mass of rigid body A (kg)} \item{${m_{B}}$ is the mass of rigid body B (kg)} - \item{$||\mathbf{n}||$ is the length of the normal vector (m)} - \item{$||{\mathbf{r}_{AP}}*\mathbf{n}||$ is the length of the perpendicular vector to the contact displacement vector of rigid body A (m)} - \item{${\mathbf{I}_{A}}$ is the moment of inertia of rigid body A (kg$\text{m}^{2}$)} - \item{$||{\mathbf{r}_{BP}}*\mathbf{n}||$ is the length of the perpendicular vector to the contact displacement vector of rigid body B (m)} - \item{${\mathbf{I}_{B}}$ is the moment of inertia of rigid body B (kg$\text{m}^{2}$)} + \item{$\text{||}\mathbf{n}\text{||}$ is the length of the normal vector (m)} + \item{$\text{||}{\mathbf{r}_{\text{A}P}}\text{*}\mathbf{n}\text{||}$ is the length of the perpendicular vector to the contact displacement vector of rigid body A (m)} + \item{${\mathbf{I}_{\text{A}}}$ is the moment of inertia of rigid body A (kg$\text{m}^{2}$)} + \item{$\text{||}{\mathbf{r}_{\text{B}P}}\text{*}\mathbf{n}\text{||}$ is the length of the perpendicular vector to the contact displacement vector of rigid body B (m)} + \item{${\mathbf{I}_{\text{B}}}$ is the moment of inertia of rigid body B (kg$\text{m}^{2}$)} \end{symbDescription} \\ \midrule \\ Notes & \hyperref[assumpOT]{A: objectTy} @@ -967,11 +967,11 @@ \subsubsection{Data Definitions} \end{displaymath} Rearranging : \begin{displaymath} -\int_{{t_{1}}}^{{t_{2}}}{\mathbf{F}}\,dt=m \left(\int_{{\mathbf{v}_{1}}}^{{\mathbf{v}_{2}}}{1}\,d\mathbf{v}\right) +\int_{{t_{\text{1}}}}^{{t_{\text{2}}}}{\mathbf{F}}\,dt=m \left(\int_{{\mathbf{v}_{\text{1}}}}^{{\mathbf{v}_{\text{2}}}}{1}\,d\mathbf{v}\right) \end{displaymath} Integrating the right hand side : \begin{displaymath} -\int_{{t_{1}}}^{{t_{2}}}{\mathbf{F}}\,dt=m {\mathbf{v}_{2}}-m {\mathbf{v}_{1}}=m Δ\mathbf{v} +\int_{{t_{\text{1}}}}^{{t_{\text{2}}}}{\mathbf{F}}\,dt=m {\mathbf{v}_{\text{2}}}-m {\mathbf{v}_{\text{1}}}=m Δ\mathbf{v} \end{displaymath} \par~ diff --git a/code/stable/glassbr/SRS/GlassBR_SRS.tex b/code/stable/glassbr/SRS/GlassBR_SRS.tex index 61caa8d532..6aa7842930 100644 --- a/code/stable/glassbr/SRS/GlassBR_SRS.tex +++ b/code/stable/glassbr/SRS/GlassBR_SRS.tex @@ -66,7 +66,7 @@ \subsection{Table of Symbols} \\ $AR$ & Aspect ratio & -- \\ -${AR_{max}}$ & Maximum aspect ratio & -- +${AR_{\text{max}}}$ & Maximum aspect ratio & -- \\ $B$ & Risk of failure & -- \\ @@ -74,9 +74,9 @@ \subsection{Table of Symbols} \\ $capacity$ & Capacity or load resistance & Pa \\ -${d_{max}}$ & Maximum value for one of the dimensions of the glass plate & m +${d_{\text{max}}}$ & Maximum value for one of the dimensions of the glass plate & m \\ -${d_{min}}$ & Minimum value for one of the dimensions of the glass plate & m +${d_{\text{min}}}$ & Minimum value for one of the dimensions of the glass plate & m \\ $E$ & Modulus of elasticity of glass & Pa \\ @@ -100,7 +100,7 @@ \subsection{Table of Symbols} \\ $J$ & Stress distribution factor (Function) & -- \\ -${J_{tol}}$ & Stress distribution factor (Function) based on Pbtol & -- +${J_{\text{tol}}}$ & Stress distribution factor (Function) based on Pbtol & -- \\ $k$ & Surface flaw parameter & $\frac{\text{m}^{12}}{\text{N}^{7}}$ \\ @@ -118,23 +118,23 @@ \subsection{Table of Symbols} \\ ${P_{b}}$ & Probability of breakage & -- \\ -${P_{btol}}$ & Tolerable probability of breakage & -- +${P_{\text{btol}}}$ & Tolerable probability of breakage & -- \\ ${P_{f}}$ & Probability of failure & -- \\ -${P_{ftol}}$ & Tolerable probability of failure & -- +${P_{\text{ftol}}}$ & Tolerable probability of failure & -- \\ $q$ & Applied load (demand) & Pa \\ $\hat{q}$ & Dimensionless load & -- \\ -${\hat{q}_{tol}}$ & Tolerable load & -- +${\hat{q}_{\text{tol}}}$ & Tolerable load & -- \\ $SD$ & Stand off distance & m \\ -${SD_{max}}$ & Maximum stand off distance permissible for input & m +${SD_{\text{max}}}$ & Maximum stand off distance permissible for input & m \\ -${SD_{min}}$ & Minimum stand off distance permissible for input & m +${SD_{\text{min}}}$ & Minimum stand off distance permissible for input & m \\ ${SD_{x}}$ & stand off distance ($x$-component) & m \\ @@ -150,9 +150,9 @@ \subsection{Table of Symbols} \\ $w$ & Charge weight & kg \\ -${w_{max}}$ & Maximum permissible input charge weight & kg +${w_{\text{max}}}$ & Maximum permissible input charge weight & kg \\ -${w_{min}}$ & Minimum permissible input charge weight & kg +${w_{\text{min}}}$ & Minimum permissible input charge weight & kg \\ ${w_{TNT}}$ & Explosive mass in equivalent weight of TNT & kg \\ @@ -368,16 +368,16 @@ \subsubsection{Theoretical Models} Label & Safety Probability \\ \midrule \\ Equation & \begin{displaymath} - is-safeProb={P_{f}}<{P_{ftol}} + is-safeProb={P_{f}}<{P_{\text{ftol}}} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} \item{$is-safeProb$ is the variable that is assigned true when probability of failure is less than tolerable probability of failure (Unitless)} \item{${P_{f}}$ is the probability of failure (Unitless)} - \item{${P_{ftol}}$ is the tolerable probability of failure (Unitless)} + \item{${P_{\text{ftol}}}$ is the tolerable probability of failure (Unitless)} \end{symbDescription} \\ \midrule \\ -Notes & If $is-safeProb$, the glass is considered safe. $is-safeProb$ and $is-safeLoad$ (from \hyperref[TM:isSafeLoad]{TM: isSafeLoad}) are either both True or both False. ${P_{f}}$ is the probability of failure, ${P_{ftol}}$ is the tolerable probability of failure. +Notes & If $is-safeProb$, the glass is considered safe. $is-safeProb$ and $is-safeLoad$ (from \hyperref[TM:isSafeLoad]{TM: isSafeLoad}) are either both True or both False. ${P_{f}}$ is the probability of failure, ${P_{\text{ftol}}}$ is the tolerable probability of failure. \\ \midrule \\ Source & \cite{astm2009} \\ \midrule \\ @@ -587,12 +587,12 @@ \subsubsection{Data Definitions} Units & Pa \\ \midrule \\ Equation & \begin{displaymath} - NFL=\frac{{\hat{q}_{tol}} E h^{4}}{\left(a b\right)^{2}} + NFL=\frac{{\hat{q}_{\text{tol}}} E h^{4}}{\left(a b\right)^{2}} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} \item{$NFL$ is the non-factored load (Pa)} - \item{${\hat{q}_{tol}}$ is the tolerable load (Unitless)} + \item{${\hat{q}_{\text{tol}}}$ is the tolerable load (Unitless)} \item{$E$ is the modulus of elasticity of glass (Pa)} \item{$h$ is the minimum thickness (m)} \item{$a$ is the plate length (long dimension) (m)} @@ -601,7 +601,7 @@ \subsubsection{Data Definitions} \\ \midrule \\ Notes & $a$ and $b$ are dimensions of the plate, where ($a\geq{}b$). $h$ is the minimum thickness, which is based on the nominal thicknesses as shown in \hyperref[DD:minThick]{DD: minThick}. - ${\hat{q}_{tol}}$ is the tolerable load defined in \hyperref[DD:tolLoad]{DD: tolLoad}. + ${\hat{q}_{\text{tol}}}$ is the tolerable load defined in \hyperref[DD:tolLoad]{DD: tolLoad}. \hyperref[assumpSV]{A: standardValues} \\ \midrule \\ Source & \cite{astm2009} @@ -698,22 +698,22 @@ \subsubsection{Data Definitions} \\ \midrule \\ Label & Tolerable load \\ \midrule \\ -Symbol & ${\hat{q}_{tol}}$ +Symbol & ${\hat{q}_{\text{tol}}}$ \\ \midrule \\ Units & Unitless \\ \midrule \\ Equation & \begin{displaymath} - {\hat{q}_{tol}}=interpY\left(\text{``SDF.txt''},AR,{J_{tol}}\right) + {\hat{q}_{\text{tol}}}=interpY\left(\text{``SDF.txt''},AR,{J_{\text{tol}}}\right) \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} - \item{${\hat{q}_{tol}}$ is the tolerable load (Unitless)} + \item{${\hat{q}_{\text{tol}}}$ is the tolerable load (Unitless)} \item{$interpY$ is the interpY (Unitless)} \item{$AR$ is the aspect ratio (Unitless)} - \item{${J_{tol}}$ is the stress distribution factor (Function) based on Pbtol (Unitless)} + \item{${J_{\text{tol}}}$ is the stress distribution factor (Function) based on Pbtol (Unitless)} \end{symbDescription} \\ \midrule \\ -Notes & ${\hat{q}_{tol}}$ is the tolerable load which is obtained from \hyperref[Figure:dimlessloadVSaspect]{Fig:dimlessloadVSaspect} using ${J_{tol}}$ and aspect ratio as parameters using interpolation. Calculations of ${J_{tol}}$ and $AR$ are defined in \hyperref[DD:sdfTol]{DD: sdfTol} and \hyperref[DD:aspectRatio]{DD: aspectRatio}, respectively. +Notes & ${\hat{q}_{\text{tol}}}$ is the tolerable load which is obtained from \hyperref[Figure:dimlessloadVSaspect]{Fig:dimlessloadVSaspect} using ${J_{\text{tol}}}$ and aspect ratio as parameters using interpolation. Calculations of ${J_{\text{tol}}}$ and $AR$ are defined in \hyperref[DD:sdfTol]{DD: sdfTol} and \hyperref[DD:aspectRatio]{DD: aspectRatio}, respectively. \\ \midrule \\ Source & \cite{astm2009} \\ \midrule \\ @@ -731,17 +731,17 @@ \subsubsection{Data Definitions} \\ \midrule \\ Label & Stress distribution factor (Function) based on Pbtol \\ \midrule \\ -Symbol & ${J_{tol}}$ +Symbol & ${J_{\text{tol}}}$ \\ \midrule \\ Units & Unitless \\ \midrule \\ Equation & \begin{displaymath} - {J_{tol}}=\ln\left(\ln\left(\frac{1}{1-{P_{btol}}}\right) \frac{\left(a b\right)^{m-1}}{k \left(E h^{2}\right)^{m} LDF}\right) + {J_{\text{tol}}}=\ln\left(\ln\left(\frac{1}{1-{P_{\text{btol}}}}\right) \frac{\left(a b\right)^{m-1}}{k \left(E h^{2}\right)^{m} LDF}\right) \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} - \item{${J_{tol}}$ is the stress distribution factor (Function) based on Pbtol (Unitless)} - \item{${P_{btol}}$ is the tolerable probability of breakage (Unitless)} + \item{${J_{\text{tol}}}$ is the stress distribution factor (Function) based on Pbtol (Unitless)} + \item{${P_{\text{btol}}}$ is the tolerable probability of breakage (Unitless)} \item{$a$ is the plate length (long dimension) (m)} \item{$b$ is the plate width (short dimension) (m)} \item{$m$ is the surface flaw parameter ($\frac{\text{m}^{12}}{\text{N}^{7}}$)} @@ -751,11 +751,11 @@ \subsubsection{Data Definitions} \item{$LDF$ is the load duration factor (Unitless)} \end{symbDescription} \\ \midrule \\ -Notes & ${J_{tol}}$ is calculated with reference to ${P_{btol}}$. +Notes & ${J_{\text{tol}}}$ is calculated with reference to ${P_{\text{btol}}}$. $a$ and $b$ are dimensions of the plate, where ($a\geq{}b$). $h$ is the minimum thickness, which is based on the nominal thicknesses as shown in \hyperref[DD:minThick]{DD: minThick}. $LDF$ is the load duration factor as defined by \hyperref[DD:loadDurFactor]{DD: loadDurFactor}. - ${P_{btol}}$ is the tolerable probability entered by the user. + ${P_{\text{btol}}}$ is the tolerable probability entered by the user. \hyperref[assumpSV]{A: standardValues} \\ \midrule \\ Source & \cite{astm2009} @@ -971,7 +971,7 @@ \subsubsection{Instance Models} \\ \midrule \\ Label & Safety Req-Pb \\ \midrule \\ -Input & ${P_{b}}$, ${P_{btol}}$ +Input & ${P_{b}}$, ${P_{\text{btol}}}$ \\ \midrule \\ Output & $is-safePb$ \\ \midrule \\ @@ -979,22 +979,22 @@ \subsubsection{Instance Models} {P_{b}}>0 \end{displaymath} \begin{displaymath} - {P_{btol}}>0 + {P_{\text{btol}}}>0 \end{displaymath} \\ \midrule \\ Output Constraints & \\ \midrule \\ Equation & \begin{displaymath} - is-safePb={P_{b}}<{P_{btol}} + is-safePb={P_{b}}<{P_{\text{btol}}} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} \item{$is-safePb$ is the variable that is assigned true when calculated probability is less than tolerable probability (Unitless)} \item{${P_{b}}$ is the probability of breakage (Unitless)} - \item{${P_{btol}}$ is the tolerable probability of breakage (Unitless)} + \item{${P_{\text{btol}}}$ is the tolerable probability of breakage (Unitless)} \end{symbDescription} \\ \midrule \\ -Notes & If $is-safePb$, the glass is considered safe. $is-safePb$ and $is-safePb$ (from \hyperref[IM:isSafeLR]{IM: isSafeLR}) are either both True or both False. ${P_{b}}$ is the probability of breakage, as calculated in \hyperref[DD:probOfBreak]{DD: probOfBreak}. ${P_{btol}}$ is the tolerable probability of breakage entered by the user. +Notes & If $is-safePb$, the glass is considered safe. $is-safePb$ and $is-safePb$ (from \hyperref[IM:isSafeLR]{IM: isSafeLR}) are either both True or both False. ${P_{b}}$ is the probability of breakage, as calculated in \hyperref[DD:probOfBreak]{DD: probOfBreak}. ${P_{\text{btol}}}$ is the tolerable probability of breakage entered by the user. \\ \midrule \\ Source & \cite{astm2009} \\ \midrule \\ @@ -1052,19 +1052,19 @@ \subsubsection{Data Constraints} \\ \midrule \endhead -$a$ & $a>0\land{}a\geq{}b$ & ${d_{min}}\leq{}a\leq{}{d_{max}}$ & $1.5$ m & 10$\%$ +$a$ & $a>0\land{}a\geq{}b$ & ${d_{\text{min}}}\leq{}a\leq{}{d_{\text{max}}}$ & $1.5$ m & 10$\%$ \\ -$AR$ & $AR\geq{}1$ & $AR\leq{}{AR_{max}}$ & $1.5$ & 10$\%$ +$AR$ & $AR\geq{}1$ & $AR\leq{}{AR_{\text{max}}}$ & $1.5$ & 10$\%$ \\ -$b$ & $00$ & ${SD_{min}}\leq{}SD\leq{}{SD_{max}}$ & $45.0$ m & 10$\%$ +$SD$ & $SD>0$ & ${SD_{\text{min}}}\leq{}SD\leq{}{SD_{\text{max}}}$ & $45.0$ m & 10$\%$ \\ $TNT$ & $TNT>0$ & -- & $1.0$ & 10$\%$ \\ -$w$ & $w>0$ & ${w_{min}}\leq{}w\leq{}{w_{max}}$ & $42.0$ kg & 10$\%$ +$w$ & $w>0$ & ${w_{\text{min}}}\leq{}w\leq{}{w_{\text{max}}}$ & $42.0$ kg & 10$\%$ \\ \bottomrule \caption{Input Data Constraints} @@ -1111,7 +1111,7 @@ \subsection{Functional Requirements} \\ $g$ & Glass type $g\in{}\{AN,FT,HS\}$ & -- \\ -${P_{btol}}$ & Tolerable probability of breakage & -- +${P_{\text{btol}}}$ & Tolerable probability of breakage & -- \\ ${SD_{x}}$ & stand off distance ($x$-component) & m \\ @@ -1179,13 +1179,13 @@ \subsection{Functional Requirements} \\ $J$ & Stress distribution factor (Function) & \hyperref[DD:stressDistFac]{DD: stressDistFac} & -- \\ -${J_{tol}}$ & Stress distribution factor (Function) based on Pbtol & \hyperref[DD:sdfTol]{DD: sdfTol} & -- +${J_{\text{tol}}}$ & Stress distribution factor (Function) based on Pbtol & \hyperref[DD:sdfTol]{DD: sdfTol} & -- \\ $NFL$ & Non-factored load & \hyperref[DD:nFL]{DD: nFL} & Pa \\ $\hat{q}$ & Dimensionless load & \hyperref[DD:dimlessLoad]{DD: dimlessLoad} & -- \\ -${\hat{q}_{tol}}$ & Tolerable load & \hyperref[DD:tolLoad]{DD: tolLoad} & -- +${\hat{q}_{\text{tol}}}$ & Tolerable load & \hyperref[DD:tolLoad]{DD: tolLoad} & -- \\ \bottomrule \caption{Required Outputs following \hyperref[outputQuants]{FR: Output-Quantities}} @@ -1401,11 +1401,11 @@ \section{Values of Auxiliary Constants} \\ \midrule \endhead -${AR_{max}}$ & maximum aspect ratio & $5.0$ & -- +${AR_{\text{max}}}$ & maximum aspect ratio & $5.0$ & -- \\ -${d_{max}}$ & maximum value for one of the dimensions of the glass plate & $5.0$ & m +${d_{\text{max}}}$ & maximum value for one of the dimensions of the glass plate & $5.0$ & m \\ -${d_{min}}$ & minimum value for one of the dimensions of the glass plate & $0.1$ & m +${d_{\text{min}}}$ & minimum value for one of the dimensions of the glass plate & $0.1$ & m \\ $E$ & modulus of elasticity of glass & $71.7\cdot{}10^{9}$ & Pa \\ @@ -1415,15 +1415,15 @@ \section{Values of Auxiliary Constants} \\ $m$ & surface flaw parameter & $7.0$ & $\frac{\text{m}^{12}}{\text{N}^{7}}$ \\ -${SD_{max}}$ & maximum stand off distance permissible for input & $130.0$ & m +${SD_{\text{max}}}$ & maximum stand off distance permissible for input & $130.0$ & m \\ -${SD_{min}}$ & minimum stand off distance permissible for input & $6.0$ & m +${SD_{\text{min}}}$ & minimum stand off distance permissible for input & $6.0$ & m \\ ${t_{d}}$ & duration of load & $3.0$ & s \\ -${w_{max}}$ & maximum permissible input charge weight & $910.0$ & kg +${w_{\text{max}}}$ & maximum permissible input charge weight & $910.0$ & kg \\ -${w_{min}}$ & minimum permissible input charge weight & $4.5$ & kg +${w_{\text{min}}}$ & minimum permissible input charge weight & $4.5$ & kg \\ \bottomrule \caption{Auxiliary Constants} diff --git a/code/stable/nopcm/SRS/NoPCM_SRS.tex b/code/stable/nopcm/SRS/NoPCM_SRS.tex index fc463783b8..e6e844212c 100644 --- a/code/stable/nopcm/SRS/NoPCM_SRS.tex +++ b/code/stable/nopcm/SRS/NoPCM_SRS.tex @@ -66,9 +66,9 @@ \subsection{Table of Symbols} \endhead ${A_{C}}$ & Heating coil surface area & $\text{m}^{2}$ \\ -${A_{in}}$ & Surface area over which heat is transferred in & $\text{m}^{2}$ +${A_{\text{in}}}$ & Surface area over which heat is transferred in & $\text{m}^{2}$ \\ -${A_{out}}$ & Surface area over which heat is transferred out & $\text{m}^{2}$ +${A_{\text{out}}}$ & Surface area over which heat is transferred out & $\text{m}^{2}$ \\ $C$ & Specific heat capacity & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ \\ @@ -100,9 +100,9 @@ \subsection{Table of Symbols} \\ ${q_{C}}$ & Heat flux into the water from the coil & $\frac{\text{W}}{\text{m}^{2}}$ \\ -${q_{in}}$ & Heat flux input & $\frac{\text{W}}{\text{m}^{2}}$ +${q_{\text{in}}}$ & Heat flux input & $\frac{\text{W}}{\text{m}^{2}}$ \\ -${q_{out}}$ & Heat flux output & $\frac{\text{W}}{\text{m}^{2}}$ +${q_{\text{out}}}$ & Heat flux output & $\frac{\text{W}}{\text{m}^{2}}$ \\ $\mathbf{q}$ & Thermal flux vector & $\frac{\text{W}}{\text{m}^{2}}$ \\ @@ -114,17 +114,17 @@ \subsection{Table of Symbols} \\ ${T_{C}}$ & Temperature of the heating coil & ${}^{\circ}$C \\ -${T_{env}}$ & Temperature of the environment & ${}^{\circ}$C +${T_{\text{env}}}$ & Temperature of the environment & ${}^{\circ}$C \\ -${T_{init}}$ & Initial temperature & ${}^{\circ}$C +${T_{\text{init}}}$ & Initial temperature & ${}^{\circ}$C \\ ${T_{W}}$ & Temperature of the water & ${}^{\circ}$C \\ $t$ & Time & s \\ -${t_{final}}$ & Final time & s +${t_{\text{final}}}$ & Final time & s \\ -${t_{step}}$ & Time step for simulation & s +${t_{\text{step}}}$ & Time step for simulation & s \\ $V$ & Volume & $\text{m}^{3}$ \\ @@ -382,7 +382,7 @@ \subsubsection{General Definitions} \item{$ΔT$ is the change in temperature (${}^{\circ}$C)} \end{symbDescription} \\ \midrule \\ -Notes & Newton's law of cooling describes convective cooling from a surface. The law is stated as: the rate of heat loss from a body is proportional to the difference in temperatures between the body and its surroundings. $\mathbf{q}\left(t\right)$ is the thermal flux ($\frac{\text{W}}{\text{m}^{2}}$). $h$ is the heat transfer coefficient, assumed independant of $T$ (\hyperref[assumpHTCC]{A: Heat-Transfer-Coeffs-Constant}) ($\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$). $ΔT\left(t\right)=T\left(t\right)-{T_{env}}\left(t\right)$ is the time-dependant thermal gradient between the environment and the object (${}^{\circ}$C). +Notes & Newton's law of cooling describes convective cooling from a surface. The law is stated as: the rate of heat loss from a body is proportional to the difference in temperatures between the body and its surroundings. $\mathbf{q}\left(t\right)$ is the thermal flux ($\frac{\text{W}}{\text{m}^{2}}$). $h$ is the heat transfer coefficient, assumed independant of $T$ (\hyperref[assumpHTCC]{A: Heat-Transfer-Coeffs-Constant}) ($\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$). $ΔT\left(t\right)=T\left(t\right)-{T_{\text{env}}}\left(t\right)$ is the time-dependant thermal gradient between the environment and the object (${}^{\circ}$C). \\ \midrule \\ Source & \cite[(pg. 8)]{incroperaEtAl2007} \\ \midrule \\ @@ -401,7 +401,7 @@ \subsubsection{General Definitions} Label & Simplified rate of change of temperature \\ \midrule \\ Equation & \begin{displaymath} - m C \frac{\,dT}{\,dt}={q_{in}} {A_{in}}-{q_{out}} {A_{out}}+g V + m C \frac{\,dT}{\,dt}={q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} @@ -409,15 +409,15 @@ \subsubsection{General Definitions} \item{$C$ is the specific heat capacity ($\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$)} \item{$t$ is the time (s)} \item{$T$ is the temperature (${}^{\circ}$C)} - \item{${q_{in}}$ is the heat flux input ($\frac{\text{W}}{\text{m}^{2}}$)} - \item{${A_{in}}$ is the surface area over which heat is transferred in ($\text{m}^{2}$)} - \item{${q_{out}}$ is the heat flux output ($\frac{\text{W}}{\text{m}^{2}}$)} - \item{${A_{out}}$ is the surface area over which heat is transferred out ($\text{m}^{2}$)} + \item{${q_{\text{in}}}$ is the heat flux input ($\frac{\text{W}}{\text{m}^{2}}$)} + \item{${A_{\text{in}}}$ is the surface area over which heat is transferred in ($\text{m}^{2}$)} + \item{${q_{\text{out}}}$ is the heat flux output ($\frac{\text{W}}{\text{m}^{2}}$)} + \item{${A_{\text{out}}}$ is the surface area over which heat is transferred out ($\text{m}^{2}$)} \item{$g$ is the volumetric heat generation per unit volume ($\frac{\text{W}}{\text{m}^{3}}$)} \item{$V$ is the volume ($\text{m}^{3}$)} \end{symbDescription} \\ \midrule \\ -Notes & The basic equation governing the rate of change of temperature, for a given volume $V$, with time. $m$ is the mass (kg). $C$ is the specific heat capacity ($\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$). $T$ is the temperature (${}^{\circ}$C) and $t$ is the time (s). ${q_{in}}$ and ${q_{out}}$ are the in and out heat transfer rates, respectively ($\frac{\text{W}}{\text{m}^{2}}$). ${A_{in}}$ and ${A_{out}}$ are the surface areas over which the heat is being transferred in and out, respectively ($\text{m}^{2}$). $g$ is the volumetric heat generated ($\frac{\text{W}}{\text{m}^{3}}$). $V$ is the volume ($\text{m}^{3}$). +Notes & The basic equation governing the rate of change of temperature, for a given volume $V$, with time. $m$ is the mass (kg). $C$ is the specific heat capacity ($\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$). $T$ is the temperature (${}^{\circ}$C) and $t$ is the time (s). ${q_{\text{in}}}$ and ${q_{\text{out}}}$ are the in and out heat transfer rates, respectively ($\frac{\text{W}}{\text{m}^{2}}$). ${A_{\text{in}}}$ and ${A_{\text{out}}}$ are the surface areas over which the heat is being transferred in and out, respectively ($\text{m}^{2}$). $g$ is the volumetric heat generated ($\frac{\text{W}}{\text{m}^{3}}$). $V$ is the volume ($\text{m}^{3}$). \\ \midrule \\ Source & -- \\ \midrule \\ @@ -437,15 +437,15 @@ \subsubsection{General Definitions} \end{displaymath} We consider an arbitrary volume. The volumetric heat generation per unit volume is assumed constant. Then (1) can be written as: \begin{displaymath} -{q_{in}} {A_{in}}-{q_{out}} {A_{out}}+g V=\int_{V}{ρ C \frac{\,\partial{}T}{\,\partial{}t}}\,dV +{q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V=\int_{V}{ρ C \frac{\,\partial{}T}{\,\partial{}t}}\,dV \end{displaymath} -Where ${q_{in}}$, ${q_{out}}$, ${A_{in}}$, and ${A_{out}}$ are explained in \hyperref[GD:rocTempSimp]{GD: rocTempSimp}. Assuming $ρ$, $C$ and $T$ are constant over the volume, which is true in our case by \hyperref[assumpCWTAT]{A: Constant-Water-Temp-Across-Tank}, \hyperref[assumpDWCoW]{A: Density-Water-Constant-over-Volume}, and \hyperref[assumpSHECoW]{A: Specific-Heat-Energy-Constant-over-Volume}, we have: +Where ${q_{\text{in}}}$, ${q_{\text{out}}}$, ${A_{\text{in}}}$, and ${A_{\text{out}}}$ are explained in \hyperref[GD:rocTempSimp]{GD: rocTempSimp}. Assuming $ρ$, $C$ and $T$ are constant over the volume, which is true in our case by \hyperref[assumpCWTAT]{A: Constant-Water-Temp-Across-Tank}, \hyperref[assumpDWCoW]{A: Density-Water-Constant-over-Volume}, and \hyperref[assumpSHECoW]{A: Specific-Heat-Energy-Constant-over-Volume}, we have: \begin{displaymath} -ρ C V \frac{\,dT}{\,dt}={q_{in}} {A_{in}}-{q_{out}} {A_{out}}+g V +ρ C V \frac{\,dT}{\,dt}={q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V \end{displaymath} Using the fact that $ρ$=$m$/$V$, (2) can be written as: \begin{displaymath} -m C \frac{\,dT}{\,dt}={q_{in}} {A_{in}}-{q_{out}} {A_{out}}+g V +m C \frac{\,dT}{\,dt}={q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V \end{displaymath} \subsubsection{Data Definitions} \label{Sec:DDs} @@ -499,16 +499,16 @@ \subsubsection{Instance Models} \\ \midrule \\ Label & Energy balance on water to find the temperature of the water \\ \midrule \\ -Input & ${T_{C}}$, ${T_{init}}$, ${t_{final}}$, ${A_{C}}$, ${h_{C}}$, ${C_{W}}$, ${m_{W}}$ +Input & ${T_{C}}$, ${T_{\text{init}}}$, ${t_{\text{final}}}$, ${A_{C}}$, ${h_{C}}$, ${C_{W}}$, ${m_{W}}$ \\ \midrule \\ Output & ${T_{W}}$ \\ \midrule \\ Input Constraints & \begin{displaymath} - {T_{init}}\leq{}{T_{C}} + {T_{\text{init}}}\leq{}{T_{C}} \end{displaymath} \\ \midrule \\ Output Constraints & \begin{displaymath} - 00$ & ${A_{C}}\leq{}{{A_{C}}^{max}}$ & $0.12$ $\text{m}^{2}$ & 10$\%$ +${A_{C}}$ & ${A_{C}}>0$ & ${A_{C}}\leq{}{{A_{C}}^{\text{max}}}$ & $0.12$ $\text{m}^{2}$ & 10$\%$ \\ -${C_{W}}$ & ${C_{W}}>0$ & ${{C_{W}}^{min}}<{C_{W}}<{{C_{W}}^{max}}$ & $4.186\cdot{}10^{3}$ $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ & 10$\%$ +${C_{W}}$ & ${C_{W}}>0$ & ${{C_{W}}^{\text{min}}}<{C_{W}}<{{C_{W}}^{\text{max}}}$ & $4.186\cdot{}10^{3}$ $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ & 10$\%$ \\ $D$ & $D>0$ & -- & $0.412$ m & 10$\%$ \\ -${h_{C}}$ & ${h_{C}}>0$ & ${{h_{C}}^{min}}\leq{}{h_{C}}\leq{}{{h_{C}}^{max}}$ & $1.0\cdot{}10^{3}$ $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ & 10$\%$ +${h_{C}}$ & ${h_{C}}>0$ & ${{h_{C}}^{\text{min}}}\leq{}{h_{C}}\leq{}{{h_{C}}^{\text{max}}}$ & $1.0\cdot{}10^{3}$ $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ & 10$\%$ \\ -$L$ & $L>0$ & ${L_{min}}\leq{}L\leq{}{L_{max}}$ & $1.5$ m & 10$\%$ +$L$ & $L>0$ & ${L_{\text{min}}}\leq{}L\leq{}{L_{\text{max}}}$ & $1.5$ m & 10$\%$ \\ ${T_{C}}$ & $0<{T_{C}}<100$ & -- & $50.0$ ${}^{\circ}$C & 10$\%$ \\ -${T_{init}}$ & $0<{T_{init}}<100$ & -- & $40.0$ ${}^{\circ}$C & 10$\%$ +${T_{\text{init}}}$ & $0<{T_{\text{init}}}<100$ & -- & $40.0$ ${}^{\circ}$C & 10$\%$ \\ -${t_{final}}$ & ${t_{final}}>0$ & ${t_{final}}<{{t_{final}}^{max}}$ & $50.0\cdot{}10^{3}$ s & 10$\%$ +${t_{\text{final}}}$ & ${t_{\text{final}}}>0$ & ${t_{\text{final}}}<{{t_{\text{final}}}^{\text{max}}}$ & $50.0\cdot{}10^{3}$ s & 10$\%$ \\ -${t_{step}}$ & $0<{t_{step}}<{t_{final}}$ & -- & $0.01$ s & 10$\%$ +${t_{\text{step}}}$ & $0<{t_{\text{step}}}<{t_{\text{final}}}$ & -- & $0.01$ s & 10$\%$ \\ -${ρ_{W}}$ & ${ρ_{W}}>0$ & ${{ρ_{W}}^{min}}<{ρ_{W}}\leq{}{{ρ_{W}}^{max}}$ & $1.0\cdot{}10^{3}$ $\frac{\text{kg}}{\text{m}^{3}}$ & 10$\%$ +${ρ_{W}}$ & ${ρ_{W}}>0$ & ${{ρ_{W}}^{\text{min}}}<{ρ_{W}}\leq{}{{ρ_{W}}^{\text{max}}}$ & $1.0\cdot{}10^{3}$ $\frac{\text{kg}}{\text{m}^{3}}$ & 10$\%$ \\ \bottomrule \caption{Input Data Constraints} @@ -631,7 +631,7 @@ \subsubsection{Properties of a Correct Solution} \\ \midrule \endhead -${T_{W}}$ & ${T_{init}}\leq{}{T_{W}}\leq{}{T_{C}}$ +${T_{W}}$ & ${T_{\text{init}}}\leq{}{T_{W}}\leq{}{T_{C}}$ \\ ${E_{W}}$ & ${E_{W}}\geq{}0$ \\ @@ -661,7 +661,7 @@ \subsection{Functional Requirements} \endhead ${A_{C}}$ & Heating coil surface area & $\text{m}^{2}$ \\ -${A_{tol}}$ & Absolute tolerance & -- +${A_{\text{tol}}}$ & Absolute tolerance & -- \\ ${C_{W}}$ & Specific heat capacity of water & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ \\ @@ -671,15 +671,15 @@ \subsection{Functional Requirements} \\ $L$ & Length of tank & m \\ -${R_{tol}}$ & Relative tolerance & -- +${R_{\text{tol}}}$ & Relative tolerance & -- \\ ${T_{C}}$ & Temperature of the heating coil & ${}^{\circ}$C \\ -${T_{init}}$ & Initial temperature & ${}^{\circ}$C +${T_{\text{init}}}$ & Initial temperature & ${}^{\circ}$C \\ -${t_{final}}$ & Final time & s +${t_{\text{final}}}$ & Final time & s \\ -${t_{step}}$ & Time step for simulation & s +${t_{\text{step}}}$ & Time step for simulation & s \\ ${ρ_{W}}$ & Density of water & $\frac{\text{kg}}{\text{m}^{3}}$ \\ @@ -843,27 +843,27 @@ \section{Values of Auxiliary Constants} \\ \midrule \endhead -${{A_{C}}^{max}}$ & maximum surface area of coil & $100000$ & $\text{m}^{2}$ +${{A_{C}}^{\text{max}}}$ & maximum surface area of coil & $100000$ & $\text{m}^{2}$ \\ -${C_{tol}}$ & relative tolerance for conservation of energy & $0.001\%$ & -- +${C_{\text{tol}}}$ & relative tolerance for conservation of energy & $0.001\%$ & -- \\ -${{C_{W}}^{max}}$ & maximum specific heat capacity of water & $4210$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ +${{C_{W}}^{\text{max}}}$ & maximum specific heat capacity of water & $4210$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ \\ -${{C_{W}}^{min}}$ & minimum specific heat capacity of water & $4170$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ +${{C_{W}}^{\text{min}}}$ & minimum specific heat capacity of water & $4170$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ \\ -${{h_{C}}^{max}}$ & maximum convective heat transfer coefficient between coil and water & $10000$ & $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ +${{h_{C}}^{\text{max}}}$ & maximum convective heat transfer coefficient between coil and water & $10000$ & $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ \\ -${{h_{C}}^{min}}$ & minimum convective heat transfer coefficient between coil and water & $10$ & $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ +${{h_{C}}^{\text{min}}}$ & minimum convective heat transfer coefficient between coil and water & $10$ & $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ \\ -${L_{max}}$ & maximum length of tank & $50$ & m +${L_{\text{max}}}$ & maximum length of tank & $50$ & m \\ -${L_{min}}$ & minimum length of tank & $0.1$ & m +${L_{\text{min}}}$ & minimum length of tank & $0.1$ & m \\ -${{t_{final}}^{max}}$ & maximum final time & $86400$ & s +${{t_{\text{final}}}^{\text{max}}}$ & maximum final time & $86400$ & s \\ -${{ρ_{W}}^{max}}$ & maximum density of water & $1000$ & $\frac{\text{kg}}{\text{m}^{3}}$ +${{ρ_{W}}^{\text{max}}}$ & maximum density of water & $1000$ & $\frac{\text{kg}}{\text{m}^{3}}$ \\ -${{ρ_{W}}^{min}}$ & minimum density of water & $950$ & $\frac{\text{kg}}{\text{m}^{3}}$ +${{ρ_{W}}^{\text{min}}}$ & minimum density of water & $950$ & $\frac{\text{kg}}{\text{m}^{3}}$ \\ \bottomrule \caption{Auxiliary Constants} diff --git a/code/stable/projectile/SRS/Projectile_SRS.tex b/code/stable/projectile/SRS/Projectile_SRS.tex index 029b3d89e2..cafca87319 100644 --- a/code/stable/projectile/SRS/Projectile_SRS.tex +++ b/code/stable/projectile/SRS/Projectile_SRS.tex @@ -74,7 +74,7 @@ \subsection{Table of Symbols} \\ ${\mathbf{a}^{c}}$ & Constant acceleration vector & $\frac{\text{m}}{\text{s}^{2}}$ \\ -${d_{offset}}$ & Distance between the target position and the landing position & m +${d_{\text{offset}}}$ & Distance between the target position and the landing position & m \\ $g$ & Gravitational acceleration & $\frac{\text{m}}{\text{s}^{2}}$ \\ @@ -82,9 +82,9 @@ \subsection{Table of Symbols} \\ ${p^{i}}$ & Initial position & m \\ -${p_{land}}$ & Landing position & m +${p_{\text{land}}}$ & Landing position & m \\ -${p_{target}}$ & Target position & m +${p_{\text{target}}}$ & Target position & m \\ ${p_{x}}$ & $x$-component of position & m \\ @@ -100,13 +100,13 @@ \subsection{Table of Symbols} \\ $t$ & Time & s \\ -${t_{flight}}$ & Flight duration & s +${t_{\text{flight}}}$ & Flight duration & s \\ $v$ & Speed & $\frac{\text{m}}{\text{s}}$ \\ ${v^{i}}$ & Initial speed & $\frac{\text{m}}{\text{s}}$ \\ -${v_{launch}}$ & Launch speed & $\frac{\text{m}}{\text{s}}$ +${v_{\text{launch}}}$ & Launch speed & $\frac{\text{m}}{\text{s}}$ \\ ${v_{x}}$ & $x$-component of velocity & $\frac{\text{m}}{\text{s}}$ \\ @@ -610,35 +610,35 @@ \subsubsection{Instance Models} \\ \midrule \\ Label & Calculation of landing time \\ \midrule \\ -Input & ${v_{launch}}$, $θ$ +Input & ${v_{\text{launch}}}$, $θ$ \\ \midrule \\ -Output & ${t_{flight}}$ +Output & ${t_{\text{flight}}}$ \\ \midrule \\ Input Constraints & \begin{displaymath} - {v_{launch}}>0 + {v_{\text{launch}}}>0 \end{displaymath} \begin{displaymath} 0<θ<\frac{π}{2} \end{displaymath} \\ \midrule \\ Output Constraints & \begin{displaymath} - {t_{flight}}>0 + {t_{\text{flight}}}>0 \end{displaymath} \\ \midrule \\ Equation & \begin{displaymath} - {t_{flight}}=\frac{2 {v_{launch}} \sin\left(θ\right)}{g} + {t_{\text{flight}}}=\frac{2 {v_{\text{launch}}} \sin\left(θ\right)}{g} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} - \item{${t_{flight}}$ is the flight duration (s)} - \item{${v_{launch}}$ is the launch speed ($\frac{\text{m}}{\text{s}}$)} + \item{${t_{\text{flight}}}$ is the flight duration (s)} + \item{${v_{\text{launch}}}$ is the launch speed ($\frac{\text{m}}{\text{s}}$)} \item{$θ$ is the launch angle (rad)} \item{$g$ is the gravitational acceleration ($\frac{\text{m}}{\text{s}^{2}}$)} \end{symbDescription} \\ \midrule \\ Notes & The constraint $0<θ<\frac{π}{2}$ is from \hyperref[posXDirection]{A: posXDirection} and \hyperref[yAxisGravity]{A: yAxisGravity}, and is shown in \hyperref[Figure:Launch]{Fig:Launch}. $g$ is defined in \hyperref[Sec:AuxConstants]{Section: Values of Auxiliary Constants}. - The constraint ${t_{flight}}>0$ is from \hyperref[timeStartZero]{A: timeStartZero}. + The constraint ${t_{\text{flight}}}>0$ is from \hyperref[timeStartZero]{A: timeStartZero}. \\ \midrule \\ Source & -- \\ \midrule \\ @@ -652,21 +652,21 @@ \subsubsection{Instance Models} \begin{displaymath} {p_{y}}={{v_{y}}^{i}} t-\frac{g t^{2}}{2} \end{displaymath} -To find the time that the projectile lands, we want to find the $t$ value (${t_{flight}}$) where ${p_{y}}=0$ (since the target is on the $x$-axis from \hyperref[targetXAxis]{A: targetXAxis}). From the equation above we get: +To find the time that the projectile lands, we want to find the $t$ value (${t_{\text{flight}}}$) where ${p_{y}}=0$ (since the target is on the $x$-axis from \hyperref[targetXAxis]{A: targetXAxis}). From the equation above we get: \begin{displaymath} -{{v_{y}}^{i}} {t_{flight}}-\frac{g {t_{flight}}^{2}}{2}=0 +{{v_{y}}^{i}} {t_{\text{flight}}}-\frac{g {t_{\text{flight}}}^{2}}{2}=0 \end{displaymath} -Dividing by ${t_{flight}}$ (with the constraint ${t_{flight}}>0$) gives us: +Dividing by ${t_{\text{flight}}}$ (with the constraint ${t_{\text{flight}}}>0$) gives us: \begin{displaymath} -{{v_{y}}^{i}}-\frac{g {t_{flight}}}{2}=0 +{{v_{y}}^{i}}-\frac{g {t_{\text{flight}}}}{2}=0 \end{displaymath} -Solving for ${t_{flight}}$ gives us: +Solving for ${t_{\text{flight}}}$ gives us: \begin{displaymath} -{t_{flight}}=\frac{2 {{v_{y}}^{i}}}{g} +{t_{\text{flight}}}=\frac{2 {{v_{y}}^{i}}}{g} \end{displaymath} -From \hyperref[DD:speedIY]{DD: speedIY} (with ${v^{i}}={v_{launch}}$) we can replace ${{v_{y}}^{i}}$: +From \hyperref[DD:speedIY]{DD: speedIY} (with ${v^{i}}={v_{\text{launch}}}$) we can replace ${{v_{y}}^{i}}$: \begin{displaymath} -{t_{flight}}=\frac{2 {v_{launch}} \sin\left(θ\right)}{g} +{t_{\text{flight}}}=\frac{2 {v_{\text{launch}}} \sin\left(θ\right)}{g} \end{displaymath} \par~ @@ -678,35 +678,35 @@ \subsubsection{Instance Models} \\ \midrule \\ Label & Calculation of landing position \\ \midrule \\ -Input & ${v_{launch}}$, $θ$ +Input & ${v_{\text{launch}}}$, $θ$ \\ \midrule \\ -Output & ${p_{land}}$ +Output & ${p_{\text{land}}}$ \\ \midrule \\ Input Constraints & \begin{displaymath} - {v_{launch}}>0 + {v_{\text{launch}}}>0 \end{displaymath} \begin{displaymath} 0<θ<\frac{π}{2} \end{displaymath} \\ \midrule \\ Output Constraints & \begin{displaymath} - {p_{land}}>0 + {p_{\text{land}}}>0 \end{displaymath} \\ \midrule \\ Equation & \begin{displaymath} - {p_{land}}=\frac{2 {v_{launch}}^{2} \sin\left(θ\right) \cos\left(θ\right)}{g} + {p_{\text{land}}}=\frac{2 {v_{\text{launch}}}^{2} \sin\left(θ\right) \cos\left(θ\right)}{g} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} - \item{${p_{land}}$ is the landing position (m)} - \item{${v_{launch}}$ is the launch speed ($\frac{\text{m}}{\text{s}}$)} + \item{${p_{\text{land}}}$ is the landing position (m)} + \item{${v_{\text{launch}}}$ is the launch speed ($\frac{\text{m}}{\text{s}}$)} \item{$θ$ is the launch angle (rad)} \item{$g$ is the gravitational acceleration ($\frac{\text{m}}{\text{s}^{2}}$)} \end{symbDescription} \\ \midrule \\ Notes & The constraint $0<θ<\frac{π}{2}$ is from \hyperref[posXDirection]{A: posXDirection} and \hyperref[yAxisGravity]{A: yAxisGravity}, and is shown in \hyperref[Figure:Launch]{Fig:Launch}. $g$ is defined in \hyperref[Sec:AuxConstants]{Section: Values of Auxiliary Constants}. - The constraint ${p_{land}}>0$ is from \hyperref[posXDirection]{A: posXDirection}. + The constraint ${p_{\text{land}}}>0$ is from \hyperref[posXDirection]{A: posXDirection}. \\ \midrule \\ Source & -- \\ \midrule \\ @@ -720,17 +720,17 @@ \subsubsection{Instance Models} \begin{displaymath} {p_{x}}={{v_{x}}^{i}} t \end{displaymath} -To find the landing position, we want to find the ${p_{x}}$ value (${p_{land}}$) at flight duration (from \hyperref[IM:calOfLandingTime]{IM: calOfLandingTime}): +To find the landing position, we want to find the ${p_{x}}$ value (${p_{\text{land}}}$) at flight duration (from \hyperref[IM:calOfLandingTime]{IM: calOfLandingTime}): \begin{displaymath} -{p_{land}}=\frac{{{v_{x}}^{i}}\cdot{}2 {v_{launch}} \sin\left(θ\right)}{g} +{p_{\text{land}}}=\frac{{{v_{x}}^{i}}\cdot{}2 {v_{\text{launch}}} \sin\left(θ\right)}{g} \end{displaymath} -From \hyperref[DD:speedIX]{DD: speedIX} (with ${v^{i}}={v_{launch}}$) we can replace ${{v_{x}}^{i}}$: +From \hyperref[DD:speedIX]{DD: speedIX} (with ${v^{i}}={v_{\text{launch}}}$) we can replace ${{v_{x}}^{i}}$: \begin{displaymath} -{p_{land}}=\frac{{v_{launch}} \cos\left(θ\right)\cdot{}2 {v_{launch}} \sin\left(θ\right)}{g} +{p_{\text{land}}}=\frac{{v_{\text{launch}}} \cos\left(θ\right)\cdot{}2 {v_{\text{launch}}} \sin\left(θ\right)}{g} \end{displaymath} Rearranging this gives us the required equation: \begin{displaymath} -{p_{land}}=\frac{2 {v_{launch}}^{2} \sin\left(θ\right) \cos\left(θ\right)}{g} +{p_{\text{land}}}=\frac{2 {v_{\text{launch}}}^{2} \sin\left(θ\right) \cos\left(θ\right)}{g} \end{displaymath} \par~ @@ -742,31 +742,31 @@ \subsubsection{Instance Models} \\ \midrule \\ Label & Offset \\ \midrule \\ -Input & ${p_{land}}$, ${p_{target}}$ +Input & ${p_{\text{land}}}$, ${p_{\text{target}}}$ \\ \midrule \\ -Output & ${d_{offset}}$ +Output & ${d_{\text{offset}}}$ \\ \midrule \\ Input Constraints & \begin{displaymath} - {p_{land}}>0 + {p_{\text{land}}}>0 \end{displaymath} \begin{displaymath} - {p_{target}}>0 + {p_{\text{target}}}>0 \end{displaymath} \\ \midrule \\ Output Constraints & \\ \midrule \\ Equation & \begin{displaymath} - {d_{offset}}={p_{land}}-{p_{target}} + {d_{\text{offset}}}={p_{\text{land}}}-{p_{\text{target}}} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} - \item{${d_{offset}}$ is the distance between the target position and the landing position (m)} - \item{${p_{land}}$ is the landing position (m)} - \item{${p_{target}}$ is the target position (m)} + \item{${d_{\text{offset}}}$ is the distance between the target position and the landing position (m)} + \item{${p_{\text{land}}}$ is the landing position (m)} + \item{${p_{\text{target}}}$ is the target position (m)} \end{symbDescription} \\ \midrule \\ -Notes & ${p_{land}}$ is from \hyperref[IM:calOfLandingDist]{IM: calOfLandingDist}. - The constraints ${p_{land}}>0$ and ${p_{target}}>0$ are from \hyperref[posXDirection]{A: posXDirection}. +Notes & ${p_{\text{land}}}$ is from \hyperref[IM:calOfLandingDist]{IM: calOfLandingDist}. + The constraints ${p_{\text{land}}}>0$ and ${p_{\text{target}}}>0$ are from \hyperref[posXDirection]{A: posXDirection}. \\ \midrule \\ Source & -- \\ \midrule \\ @@ -784,37 +784,37 @@ \subsubsection{Instance Models} \\ \midrule \\ Label & Output message \\ \midrule \\ -Input & ${d_{offset}}$, ${p_{target}}$ +Input & ${d_{\text{offset}}}$, ${p_{\text{target}}}$ \\ \midrule \\ Output & $s$ \\ \midrule \\ Input Constraints & \begin{displaymath} - {p_{target}}>0 + {p_{\text{target}}}>0 \end{displaymath} \begin{displaymath} - {d_{offset}}>-{p_{land}} + {d_{\text{offset}}}>-{p_{\text{land}}} \end{displaymath} \\ \midrule \\ Output Constraints & \\ \midrule \\ Equation & \begin{displaymath} s=\begin{cases} - \text{``The target was hit.''}, & |\frac{{d_{offset}}}{{p_{target}}}|<ε\\ -\text{``The projectile fell short.''}, & {d_{offset}}<0\\ -\text{``The projectile went long.''}, & {d_{offset}}>0 + \text{``The target was hit.''}, & |\frac{{d_{\text{offset}}}}{{p_{\text{target}}}}|<ε\\ +\text{``The projectile fell short.''}, & {d_{\text{offset}}}<0\\ +\text{``The projectile went long.''}, & {d_{\text{offset}}}>0 \end{cases} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} \item{$s$ is the output message as a string (Unitless)} - \item{${d_{offset}}$ is the distance between the target position and the landing position (m)} - \item{${p_{target}}$ is the target position (m)} + \item{${d_{\text{offset}}}$ is the distance between the target position and the landing position (m)} + \item{${p_{\text{target}}}$ is the target position (m)} \item{$ε$ is the hit tolerance (Unitless)} \end{symbDescription} \\ \midrule \\ -Notes & ${d_{offset}}$ is from \hyperref[IM:offsetIM]{IM: offsetIM}. - The constraint ${p_{target}}>0$ is from \hyperref[posXDirection]{A: posXDirection}. - The constraint ${d_{offset}}>-{p_{land}}$ is from the fact that ${p_{land}}>0$, from \hyperref[posXDirection]{A: posXDirection}. +Notes & ${d_{\text{offset}}}$ is from \hyperref[IM:offsetIM]{IM: offsetIM}. + The constraint ${p_{\text{target}}}>0$ is from \hyperref[posXDirection]{A: posXDirection}. + The constraint ${d_{\text{offset}}}>-{p_{\text{land}}}$ is from the fact that ${p_{\text{land}}}>0$, from \hyperref[posXDirection]{A: posXDirection}. $ε$ is defined in \hyperref[Sec:AuxConstants]{Section: Values of Auxiliary Constants}. \\ \midrule \\ Source & -- @@ -832,9 +832,9 @@ \subsubsection{Data Constraints} \\ \midrule \endhead -${p_{target}}$ & ${p_{target}}>0$ & $1000$ m & 10$\%$ +${p_{\text{target}}}$ & ${p_{\text{target}}}>0$ & $1000$ m & 10$\%$ \\ -${v_{launch}}$ & ${v_{launch}}>0$ & $100$ $\frac{\text{m}}{\text{s}}$ & 10$\%$ +${v_{\text{launch}}}$ & ${v_{\text{launch}}}>0$ & $100$ $\frac{\text{m}}{\text{s}}$ & 10$\%$ \\ $θ$ & $0<θ<\frac{π}{2}$ & $\frac{π}{4}$ rad & 10$\%$ \\ @@ -851,9 +851,9 @@ \subsubsection{Properties of a Correct Solution} \\ \midrule \endhead -${p_{land}}$ & ${p_{land}}>0$ +${p_{\text{land}}}$ & ${p_{\text{land}}}>0$ \\ -${d_{offset}}$ & ${d_{offset}}>-{p_{land}}$ +${d_{\text{offset}}}$ & ${d_{\text{offset}}}>-{p_{\text{land}}}$ \\ \bottomrule \caption{Output Data Constraints} @@ -868,8 +868,8 @@ \subsection{Functional Requirements} \begin{itemize} \item[Input-Parameters:\phantomsection\label{inputParams}]Input the quantities from \hyperref[Table:ReqInputs]{Table:ReqInputs}, which define the launch angle, launch speed, and target position. \item[Verify-Parameters:\phantomsection\label{verifyParams}]Check the entered input parameters to ensure that they do not exceed the data constraints mentioned in \hyperref[Sec:DataConstraints]{Section: Data Constraints}. If any of the input parameters are out of bounds, an error message is displayed and the calculations stop. -\item[Calculate-Values:\phantomsection\label{calcValues}]Calculate the following quantities: ${t_{flight}}$ (from \hyperref[IM:calOfLandingTime]{IM: calOfLandingTime}), ${p_{land}}$ (from \hyperref[IM:calOfLandingDist]{IM: calOfLandingDist}), ${d_{offset}}$ (from \hyperref[IM:offsetIM]{IM: offsetIM}), and $s$ (from \hyperref[IM:messageIM]{IM: messageIM}). -\item[Output-Values:\phantomsection\label{outputValues}]Output $s$ (from \hyperref[IM:messageIM]{IM: messageIM}) and ${d_{offset}}$ (from \hyperref[IM:offsetIM]{IM: offsetIM}). +\item[Calculate-Values:\phantomsection\label{calcValues}]Calculate the following quantities: ${t_{\text{flight}}}$ (from \hyperref[IM:calOfLandingTime]{IM: calOfLandingTime}), ${p_{\text{land}}}$ (from \hyperref[IM:calOfLandingDist]{IM: calOfLandingDist}), ${d_{\text{offset}}}$ (from \hyperref[IM:offsetIM]{IM: offsetIM}), and $s$ (from \hyperref[IM:messageIM]{IM: messageIM}). +\item[Output-Values:\phantomsection\label{outputValues}]Output $s$ (from \hyperref[IM:messageIM]{IM: messageIM}) and ${d_{\text{offset}}}$ (from \hyperref[IM:offsetIM]{IM: offsetIM}). \end{itemize} \begin{longtable}{l l l} \toprule @@ -877,9 +877,9 @@ \subsection{Functional Requirements} \\ \midrule \endhead -${p_{target}}$ & Target position & m +${p_{\text{target}}}$ & Target position & m \\ -${v_{launch}}$ & Launch speed & $\frac{\text{m}}{\text{s}}$ +${v_{\text{launch}}}$ & Launch speed & $\frac{\text{m}}{\text{s}}$ \\ $θ$ & Launch angle & rad \\ diff --git a/code/stable/ssp/SRS/SSP_SRS.tex b/code/stable/ssp/SRS/SSP_SRS.tex index 908dd74405..240fe26127 100644 --- a/code/stable/ssp/SRS/SSP_SRS.tex +++ b/code/stable/ssp/SRS/SSP_SRS.tex @@ -64,7 +64,7 @@ \subsection{Table of Symbols} \\ \midrule \endhead -$(x,y)$ & Cartesian Position Coordinates: y is considered parallel to the direction of the force of gravity and x is considered perpendicular to y & m +$\text{(x,y)}$ & Cartesian Position Coordinates: y is considered parallel to the direction of the force of gravity and x is considered perpendicular to y & m \\ $A$ & Area: A part of an object or surface & $\text{m}^{2}$ \\ @@ -72,19 +72,19 @@ \subsection{Table of Symbols} \\ $\mathbf{b}$ & Base Width of Slices: in the x-direction & m \\ -${\mathbf{C}_{den}}$ & Proportionality Constant Denominator: values for each slice that sum together to form the denominator of the interslice normal to shear force proportionality constant & N +${\mathbf{C}_{\text{den}}}$ & Proportionality Constant Denominator: values for each slice that sum together to form the denominator of the interslice normal to shear force proportionality constant & N \\ -${\mathbf{C}_{num}}$ & Proportionality Constant Numerator: values for each slice that sum together to form the numerator of the interslice normal to shear force proportionality constant & N +${\mathbf{C}_{\text{num}}}$ & Proportionality Constant Numerator: values for each slice that sum together to form the numerator of the interslice normal to shear force proportionality constant & N \\ $c'$ & Effective Cohesion: internal pressure that sticks particles of soil together & Pa \\ $const_f$ & Decision on F: boolean decision on which form of f the user desires: constant if true, or half-sine if false & -- \\ -${F_{rot}}$ & Force Causing Rotation: a force in the direction of rotation & N +${F_{\text{rot}}}$ & Force Causing Rotation: a force in the direction of rotation & N \\ -${F_{S}}$ & Factor of Safety: The global stability metric of a slip surface of a slope, defined as the ratio of resistive shear force to mobilized shear force. & -- +${F_{\text{S}}}$ & Factor of Safety: The global stability metric of a slip surface of a slope, defined as the ratio of resistive shear force to mobilized shear force. & -- \\ -${{F_{S}}^{min}}$ & Minimum Factor of Safety: The minimum factor of safety associated with the critical slip surface & -- +${{F_{\text{S}}}^{\text{min}}}$ & Minimum Factor of Safety: The minimum factor of safety associated with the critical slip surface & -- \\ ${F_{x}}$ & X-Coordinate of the Force: & N \\ @@ -114,7 +114,7 @@ \subsection{Table of Symbols} \\ ${\mathbf{h}_{z}}$ & Heights of Interslice Normal Forces: the heights in the y-direction of the interslice normal forces on each slice & m \\ -${\mathbf{h}_{z,w}}$ & Heights of the Water Table: the heights in the y-direction from the base of each slice to the watertable & m +${\mathbf{h}_{\text{z,w}}}$ & Heights of the Water Table: the heights in the y-direction from the base of each slice to the watertable & m \\ $i$ & Index: representing a single slice & -- \\ @@ -160,9 +160,9 @@ \subsection{Table of Symbols} \\ $V$ & Volume: the amount of space that a substance or object occupies. & $\text{m}^{3}$ \\ -${\mathbf{V}_{dry}}$ & Volumes of Dry Soil: amount of space occupied by dry soil for each slice & $\text{m}^{3}$ +${\mathbf{V}_{\text{dry}}}$ & Volumes of Dry Soil: amount of space occupied by dry soil for each slice & $\text{m}^{3}$ \\ -${\mathbf{V}_{sat}}$ & Volumes of Saturated Soil: amount of space occupied by saturated soil for each slice & $\text{m}^{3}$ +${\mathbf{V}_{\text{sat}}}$ & Volumes of Saturated Soil: amount of space occupied by saturated soil for each slice & $\text{m}^{3}$ \\ $v$ & Local Index: used as a bound variable index in calculations & -- \\ @@ -174,33 +174,33 @@ \subsection{Table of Symbols} \\ $x$ & X-Coordinate: in the Cartesian coordinate system & m \\ -${{x_{slip}}^{maxEtr}}$ & Maximum Entry X-Coordinate: maximum potential x-coordinate for the entry point of a slip surface & m +${{x_{\text{slip}}}^{\text{maxEtr}}}$ & Maximum Entry X-Coordinate: maximum potential x-coordinate for the entry point of a slip surface & m \\ -${{x_{slip}}^{maxExt}}$ & Maximum Exit X-Coordinate: maximum potential x-coordinate for the exit point of a slip surface & m +${{x_{\text{slip}}}^{\text{maxExt}}}$ & Maximum Exit X-Coordinate: maximum potential x-coordinate for the exit point of a slip surface & m \\ -${{x_{slip}}^{minEtr}}$ & Minimum Exit X-Coordinate: minimum potential x-coordinate for the entry point of a slip surface & m +${{x_{\text{slip}}}^{\text{minEtr}}}$ & Minimum Exit X-Coordinate: minimum potential x-coordinate for the entry point of a slip surface & m \\ -${{x_{slip}}^{minExt}}$ & Minimum Exit X-Coordinate: minimum potential x-coordinate for the exit point of a slip surface & m +${{x_{\text{slip}}}^{\text{minExt}}}$ & Minimum Exit X-Coordinate: minimum potential x-coordinate for the exit point of a slip surface & m \\ -${\mathbf{x}_{cs}},{\mathbf{y}_{cs}}$ & The Set of X and Y Coordinates: describe the vertices of the critical slip surface & m +${\mathbf{x}_{\text{cs}}}\text{,}{\mathbf{y}_{\text{cs}}}$ & The Set of X and Y Coordinates: describe the vertices of the critical slip surface & m \\ -${\mathbf{x}_{slip}}$ & X-Coordinates of the Slip Surface: x-coordinates of points on the slip surface & m +${\mathbf{x}_{\text{slip}}}$ & X-Coordinates of the Slip Surface: x-coordinates of points on the slip surface & m \\ -${\mathbf{x}_{slope}}$ & X-Coordinates of the Slope: x-coordinates of points on the soil slope & m +${\mathbf{x}_{\text{slope}}}$ & X-Coordinates of the Slope: x-coordinates of points on the soil slope & m \\ -${\mathbf{x}_{wt}}$ & X-Coordinates of the Water Table: x-positions of the water table & m +${\mathbf{x}_{\text{wt}}}$ & X-Coordinates of the Water Table: x-positions of the water table & m \\ $y$ & Y-Coordinate: in the Cartesian coordinate system & m \\ -${{y_{slip}}^{max}}$ & Maximum Y-Coordinate: maximum potential y-coordinate of a point on a slip surface & m +${{y_{\text{slip}}}^{\text{max}}}$ & Maximum Y-Coordinate: maximum potential y-coordinate of a point on a slip surface & m \\ -${{y_{slip}}^{min}}$ & Minimum Y-Coordinate: minimum potential y-coordinate of a point on a slip surface & m +${{y_{\text{slip}}}^{\text{min}}}$ & Minimum Y-Coordinate: minimum potential y-coordinate of a point on a slip surface & m \\ -${\mathbf{y}_{slip}}$ & Y-Coordinates of the Slip Surface: heights of the slip surface & m +${\mathbf{y}_{\text{slip}}}$ & Y-Coordinates of the Slip Surface: heights of the slip surface & m \\ -${\mathbf{y}_{slope}}$ & Y-Coordinates of the Slope: y-coordinates of points on the soil slope & m +${\mathbf{y}_{\text{slope}}}$ & Y-Coordinates of the Slope: y-coordinates of points on the soil slope & m \\ -${\mathbf{y}_{wt}}$ & Y-Coordinates of the Water Table: heights of the water table & m +${\mathbf{y}_{\text{wt}}}$ & Y-Coordinates of the Water Table: heights of the water table & m \\ $z$ & Z-Coordinate: in the Cartesian coordinate system & m \\ @@ -210,9 +210,9 @@ \subsection{Table of Symbols} \\ $γ$ & Specific Weight: weight per unit volume & $\frac{\text{N}}{\text{m}^{3}}$ \\ -${γ_{dry}}$ & Soil Dry Unit Weight: The weight of a dry soil/ground layer divided by the volume of the layer. & $\frac{\text{N}}{\text{m}^{3}}$ +${γ_{\text{dry}}}$ & Soil Dry Unit Weight: The weight of a dry soil/ground layer divided by the volume of the layer. & $\frac{\text{N}}{\text{m}^{3}}$ \\ -${γ_{Sat}}$ & Soil Saturated Unit Weight: The weight of saturated soil/ground layer divided by the volume of the layer. & $\frac{\text{N}}{\text{m}^{3}}$ +${γ_{\text{Sat}}}$ & Soil Saturated Unit Weight: The weight of saturated soil/ground layer divided by the volume of the layer. & $\frac{\text{N}}{\text{m}^{3}}$ \\ ${γ_{w}}$ & Unit Weight of Water: The weight of one cubic meter of water. & $\frac{\text{N}}{\text{m}^{3}}$ \\ @@ -416,7 +416,7 @@ \subsubsection{Assumptions} \label{Sec:Assumps} This section simplifies the original problem and helps in developing the theoretical models by filling in the missing information for the physical system. The assumptions refine the scope by providing more detail. \begin{itemize} -\item[Slip-Surface-Concave:\phantomsection\label{assumpSSC}]The slip surface is concave with respect to the slope surface. The (${\mathbf{x}_{slip}}$, ${\mathbf{y}_{slip}}$) coordinates of a slip surface follow a concave up function. (RefBy: \hyperref[IM:crtSlpId]{IM: crtSlpId}.) +\item[Slip-Surface-Concave:\phantomsection\label{assumpSSC}]The slip surface is concave with respect to the slope surface. The (${\mathbf{x}_{\text{slip}}}$, ${\mathbf{y}_{\text{slip}}}$) coordinates of a slip surface follow a concave up function. (RefBy: \hyperref[IM:crtSlpId]{IM: crtSlpId}.) \item[Factor-of-Safety:\phantomsection\label{assumpFOS}]The factor of safety is assumed to be constant across the entire slip surface. (RefBy: \hyperref[GD:mobShr]{GD: mobShr}.) \item[Soil-Layer-Homogeneous:\phantomsection\label{assumpSLH}]The soil mass is homogeneous, with consistent soil properties throughout. (RefBy: \hyperref[GD:sliceWght]{GD: sliceWght}, \hyperref[GD:resShr]{GD: resShr}, and \hyperref[LC_inhomogeneous]{LC: Calculate-Inhomogeneous-Soil-Layers}.) \item[Soil-Properties:\phantomsection\label{assumpSP}]The soil properties are independent of dry or saturated conditions, with the exception of unit weight. (RefBy: \hyperref[GD:resShr]{GD: resShr}.) @@ -447,11 +447,11 @@ \subsubsection{Theoretical Models} Label & Factor of safety \\ \midrule \\ Equation & \begin{displaymath} - {F_{S}}=\frac{P}{S} + {F_{\text{S}}}=\frac{P}{S} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} - \item{${F_{S}}$ is the factor of safety (Unitless)} + \item{${F_{\text{S}}}$ is the factor of safety (Unitless)} \item{$P$ is the resistive shear force (N)} \item{$S$ is the mobilized shear force (N)} \end{symbDescription} @@ -711,14 +711,14 @@ \subsubsection{General Definitions} Units & $\frac{\text{N}}{\text{m}}$ \\ \midrule \\ Equation & \begin{displaymath} - {\mathbf{S}}_{i}=\frac{{\mathbf{P}}_{i}}{{F_{S}}}=\frac{{\mathbf{N'}}_{i} \tan\left({φ'}_{i}\right)+{c'}_{i} {\mathbf{ℓ}_{b,i}}}{{F_{S}}} + {\mathbf{S}}_{i}=\frac{{\mathbf{P}}_{i}}{{F_{\text{S}}}}=\frac{{\mathbf{N'}}_{i} \tan\left({φ'}_{i}\right)+{c'}_{i} {\mathbf{ℓ}_{b,i}}}{{F_{\text{S}}}} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} \item{$\mathbf{S}$ is the mobilized shear force ($\frac{\text{N}}{\text{m}}$)} \item{$i$ is the index (Unitless)} \item{$\mathbf{P}$ is the resistive shear forces ($\frac{\text{N}}{\text{m}}$)} - \item{${F_{S}}$ is the factor of safety (Unitless)} + \item{${F_{\text{S}}}$ is the factor of safety (Unitless)} \item{$\mathbf{N'}$ is the effective normal forces ($\frac{\text{N}}{\text{m}}$)} \item{$φ'$ is the effective angle of friction (${}^{\circ}$)} \item{$c'$ is the effective cohesion (Pa)} @@ -735,7 +735,7 @@ \subsubsection{General Definitions} \end{minipage} \paragraph{} \label{GD:mobShrDeriv} -Mobilized shear forces is derived by dividing the definition of the $\mathbf{P}$ from \hyperref[GD:resShr]{GD: resShr}. by the definition of the factor of safety from \hyperref[TM:factOfSafety]{TM: factOfSafety}. The factor of safety ${F_{S}}$ is not indexed by $i$ because it is assumed to be constant for the entire slip surface (\hyperref[assumpFOS]{A: Factor-of-Safety}). +Mobilized shear forces is derived by dividing the definition of the $\mathbf{P}$ from \hyperref[GD:resShr]{GD: resShr}. by the definition of the factor of safety from \hyperref[TM:factOfSafety]{TM: factOfSafety}. The factor of safety ${F_{\text{S}}}$ is not indexed by $i$ because it is assumed to be constant for the entire slip surface (\hyperref[assumpFOS]{A: Factor-of-Safety}). \par~ \noindent \begin{minipage}{\textwidth} @@ -886,7 +886,7 @@ \subsubsection{General Definitions} Units & N \\ \midrule \\ Equation & \begin{displaymath} - 0=-{\mathbf{G}}_{i} \left({\mathbf{h}_{z,i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{G}}_{i-1} \left({\mathbf{h}_{z,i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)-{\mathbf{H}}_{i} \left(\frac{1}{3} {\mathbf{h}_{z,w,i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{H}}_{i-1} \left(\frac{1}{3} {\mathbf{h}_{z,w,i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+\frac{{\mathbf{b}}_{i}}{2} \left({\mathbf{X}}_{i}+{\mathbf{X}}_{i-1}\right)+\frac{-{K_{c}} {\mathbf{W}}_{i} {\mathbf{h}}_{i}}{2}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right) {\mathbf{h}}_{i}+{\mathbf{Q}}_{i} \sin\left({\mathbf{ω}}_{i}\right) {\mathbf{h}}_{i} + 0=-{\mathbf{G}}_{i} \left({\mathbf{h}_{z,i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{G}}_{i-1} \left({\mathbf{h}_{z,i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)-{\mathbf{H}}_{i} \left(\frac{1}{3} {\mathbf{h}_{\text{z,w},i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{H}}_{i-1} \left(\frac{1}{3} {\mathbf{h}_{\text{z,w},i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+\frac{{\mathbf{b}}_{i}}{2} \left({\mathbf{X}}_{i}+{\mathbf{X}}_{i-1}\right)+\frac{-{K_{c}} {\mathbf{W}}_{i} {\mathbf{h}}_{i}}{2}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right) {\mathbf{h}}_{i}+{\mathbf{Q}}_{i} \sin\left({\mathbf{ω}}_{i}\right) {\mathbf{h}}_{i} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} @@ -896,7 +896,7 @@ \subsubsection{General Definitions} \item{$\mathbf{b}$ is the base width of slices (m)} \item{$\mathbf{α}$ is the base angles (${}^{\circ}$)} \item{$\mathbf{H}$ is the interslice normal water forces ($\frac{\text{N}}{\text{m}}$)} - \item{${\mathbf{h}_{z,w}}$ is the heights of the water table (m)} + \item{${\mathbf{h}_{\text{z,w}}}$ is the heights of the water table (m)} \item{$\mathbf{X}$ is the interslice shear forces ($\frac{\text{N}}{\text{m}}$)} \item{${K_{c}}$ is the seismic coefficient (Unitless)} \item{$\mathbf{W}$ is the weights ($\frac{\text{N}}{\text{m}}$)} @@ -923,9 +923,9 @@ \subsubsection{General Definitions} \end{displaymath} Considering one dimension, with moments in the clockwise direction as positive and moments in the counterclockwise direction as negative, and replacing the torque symbol with the moment symbol, the equation simplifies to: \begin{displaymath} -M={F_{rot}} r +M={F_{\text{rot}}} r \end{displaymath} -where ${F_{rot}}$ is the force causing rotation and $r$ is the length of the moment arm, or the distance between the force and the axis about which the rotation acts. To represent the moment equilibrium, the moments from each force acting on a slice must be considered and added together. The forces acting on a slice are all shown in \hyperref[Figure:ForceDiagram]{Fig:ForceDiagram}. The midpoint of the base of a slice is considered as the axis of rotation, from which the length of the moment arm is measured. Considering first the interslice normal force acting on slice interface $i$, the moment is negative because the force tends to rotate the slice in a counterclockwise direction, and the length of the moment arm is the height of the force plus the difference in height between the base at slice interface $i$ and the base at the midpoint of slice $i$. Thus, the moment is expressed as: +where ${F_{\text{rot}}}$ is the force causing rotation and $r$ is the length of the moment arm, or the distance between the force and the axis about which the rotation acts. To represent the moment equilibrium, the moments from each force acting on a slice must be considered and added together. The forces acting on a slice are all shown in \hyperref[Figure:ForceDiagram]{Fig:ForceDiagram}. The midpoint of the base of a slice is considered as the axis of rotation, from which the length of the moment arm is measured. Considering first the interslice normal force acting on slice interface $i$, the moment is negative because the force tends to rotate the slice in a counterclockwise direction, and the length of the moment arm is the height of the force plus the difference in height between the base at slice interface $i$ and the base at the midpoint of slice $i$. Thus, the moment is expressed as: \begin{displaymath} -{\mathbf{G}}_{i} \left({\mathbf{h}_{z,i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right) \end{displaymath} @@ -935,11 +935,11 @@ \subsubsection{General Definitions} \end{displaymath} Next, the interslice normal water force is considered. This force is zero at the height of the water table, then increases linearly towards the base of the slice due to the increasing water pressure. For such a triangular distribution, the resultant force acts at one-third of the height. Thus, for the interslice normal water force acting on slice interface $i$, the moment is: \begin{displaymath} --{\mathbf{H}}_{i} \left(\frac{1}{3} {\mathbf{h}_{z,w,i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right) +-{\mathbf{H}}_{i} \left(\frac{1}{3} {\mathbf{h}_{\text{z,w},i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right) \end{displaymath} The moment for the interslice normal water force acting on slice interface $i-1$ is: \begin{displaymath} -{\mathbf{H}}_{i-1} \left(\frac{1}{3} {\mathbf{h}_{z,w,i-1}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right) +{\mathbf{H}}_{i-1} \left(\frac{1}{3} {\mathbf{h}_{\text{z,w},i-1}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right) \end{displaymath} The interslice shear force at slice interface $i$ tends to rotate in the clockwise direction, and the length of the moment arm is the length from the slice edge to the slice midpoint, equivalent to half of the width of the slice, so the moment is: \begin{displaymath} @@ -971,7 +971,7 @@ \subsubsection{General Definitions} \end{displaymath} The base hydrostatic force and slice weight both act in the direction of the point of rotation (\hyperref[assumpHFSM]{A: Hydrostatic-Force-Slice-Midpoint}), therefore both have moments of zero. Thus, all of the moments have been determined. The moment equilibrium is then represented by the sum of all moments: \begin{displaymath} -0=-{\mathbf{G}}_{i} \left({\mathbf{h}_{z,i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{G}}_{i-1} \left({\mathbf{h}_{z,i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)-{\mathbf{H}}_{i} \left(\frac{1}{3} {\mathbf{h}_{z,w,i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{H}}_{i-1} \left(\frac{1}{3} {\mathbf{h}_{z,w,i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+\frac{{\mathbf{b}}_{i}}{2} \left({\mathbf{X}}_{i}+{\mathbf{X}}_{i-1}\right)+\frac{-{K_{c}} {\mathbf{W}}_{i} {\mathbf{h}}_{i}}{2}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right) {\mathbf{h}}_{i}+{\mathbf{Q}}_{i} \sin\left({\mathbf{ω}}_{i}\right) {\mathbf{h}}_{i} +0=-{\mathbf{G}}_{i} \left({\mathbf{h}_{z,i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{G}}_{i-1} \left({\mathbf{h}_{z,i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)-{\mathbf{H}}_{i} \left(\frac{1}{3} {\mathbf{h}_{\text{z,w},i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{H}}_{i-1} \left(\frac{1}{3} {\mathbf{h}_{\text{z,w},i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+\frac{{\mathbf{b}}_{i}}{2} \left({\mathbf{X}}_{i}+{\mathbf{X}}_{i-1}\right)+\frac{-{K_{c}} {\mathbf{W}}_{i} {\mathbf{h}}_{i}}{2}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right) {\mathbf{h}}_{i}+{\mathbf{Q}}_{i} \sin\left({\mathbf{ω}}_{i}\right) {\mathbf{h}}_{i} \end{displaymath} \par~ @@ -1036,9 +1036,9 @@ \subsubsection{General Definitions} \\ \midrule \\ Equation & \begin{displaymath} {\mathbf{W}}_{i}={\mathbf{b}}_{i} \frac{1}{2} \begin{cases} - \left({\mathbf{y}_{slope,i}}-{\mathbf{y}_{slip,i}}+{\mathbf{y}_{slope,i-1}}-{\mathbf{y}_{slip,i-1}}\right) {γ_{Sat}}, & {\mathbf{y}_{wt,i}}>{\mathbf{y}_{slope,i}}\lor{}{\mathbf{y}_{wt,i-1}}>{\mathbf{y}_{slope,i-1}}\\ -\left({\mathbf{y}_{slope,i}}-{\mathbf{y}_{wt,i}}+{\mathbf{y}_{slope,i-1}}-{\mathbf{y}_{wt,i-1}}\right) {γ_{dry}}+\left({\mathbf{y}_{wt,i}}-{\mathbf{y}_{slip,i}}+{\mathbf{y}_{wt,i-1}}-{\mathbf{y}_{slip,i-1}}\right) {γ_{Sat}}, & {\mathbf{y}_{slope,i}}\geq{}{\mathbf{y}_{wt,i}}\geq{}{\mathbf{y}_{slip,i}}\land{}{\mathbf{y}_{slope,i-1}}\geq{}{\mathbf{y}_{wt,i-1}}\geq{}{\mathbf{y}_{slip,i-1}}\\ -\left({\mathbf{y}_{slope,i}}-{\mathbf{y}_{slip,i}}+{\mathbf{y}_{slope,i-1}}-{\mathbf{y}_{slip,i-1}}\right) {γ_{dry}}, & {\mathbf{y}_{wt,i}}<{\mathbf{y}_{slip,i}}\lor{}{\mathbf{y}_{wt,i-1}}<{\mathbf{y}_{slip,i-1}} + \left({\mathbf{y}_{\text{slope},i}}-{\mathbf{y}_{\text{slip},i}}+{\mathbf{y}_{\text{slope},i-1}}-{\mathbf{y}_{\text{slip},i-1}}\right) {γ_{\text{Sat}}}, & {\mathbf{y}_{\text{wt},i}}>{\mathbf{y}_{\text{slope},i}}\lor{}{\mathbf{y}_{\text{wt},i-1}}>{\mathbf{y}_{\text{slope},i-1}}\\ +\left({\mathbf{y}_{\text{slope},i}}-{\mathbf{y}_{\text{wt},i}}+{\mathbf{y}_{\text{slope},i-1}}-{\mathbf{y}_{\text{wt},i-1}}\right) {γ_{\text{dry}}}+\left({\mathbf{y}_{\text{wt},i}}-{\mathbf{y}_{\text{slip},i}}+{\mathbf{y}_{\text{wt},i-1}}-{\mathbf{y}_{\text{slip},i-1}}\right) {γ_{\text{Sat}}}, & {\mathbf{y}_{\text{slope},i}}\geq{}{\mathbf{y}_{\text{wt},i}}\geq{}{\mathbf{y}_{\text{slip},i}}\land{}{\mathbf{y}_{\text{slope},i-1}}\geq{}{\mathbf{y}_{\text{wt},i-1}}\geq{}{\mathbf{y}_{\text{slip},i-1}}\\ +\left({\mathbf{y}_{\text{slope},i}}-{\mathbf{y}_{\text{slip},i}}+{\mathbf{y}_{\text{slope},i-1}}-{\mathbf{y}_{\text{slip},i-1}}\right) {γ_{\text{dry}}}, & {\mathbf{y}_{\text{wt},i}}<{\mathbf{y}_{\text{slip},i}}\lor{}{\mathbf{y}_{\text{wt},i-1}}<{\mathbf{y}_{\text{slip},i-1}} \end{cases} \end{displaymath} \\ \midrule \\ @@ -1046,14 +1046,14 @@ \subsubsection{General Definitions} \item{$\mathbf{W}$ is the weights ($\frac{\text{N}}{\text{m}}$)} \item{$i$ is the index (Unitless)} \item{$\mathbf{b}$ is the base width of slices (m)} - \item{${\mathbf{y}_{slope}}$ is the y-coordinates of the slope (m)} - \item{${\mathbf{y}_{slip}}$ is the y-coordinates of the slip surface (m)} - \item{${γ_{Sat}}$ is the soil saturated unit weight ($\frac{\text{N}}{\text{m}^{3}}$)} - \item{${\mathbf{y}_{wt}}$ is the y-coordinates of the water table (m)} - \item{${γ_{dry}}$ is the soil dry unit weight ($\frac{\text{N}}{\text{m}^{3}}$)} + \item{${\mathbf{y}_{\text{slope}}}$ is the y-coordinates of the slope (m)} + \item{${\mathbf{y}_{\text{slip}}}$ is the y-coordinates of the slip surface (m)} + \item{${γ_{\text{Sat}}}$ is the soil saturated unit weight ($\frac{\text{N}}{\text{m}^{3}}$)} + \item{${\mathbf{y}_{\text{wt}}}$ is the y-coordinates of the water table (m)} + \item{${γ_{\text{dry}}}$ is the soil dry unit weight ($\frac{\text{N}}{\text{m}^{3}}$)} \end{symbDescription} \\ \midrule \\ -Notes & This equation is based on the assumption that the surface and the base of a slice are straight lines (\hyperref[assumpSBSBISL]{A: Surface-Base-Slice-between-Interslice-Straight-Lines}). The soil dry unit weight ${γ_{dry}}$ and the soil saturated unit weight ${γ_{Sat}}$ are not indexed by $i$ because the soil is assumed to be homogeneous, with constant soil properties throughout (\hyperref[assumpSLH]{A: Soil-Layer-Homogeneous}). $\mathbf{b}$ is defined in \hyperref[DD:lengthB]{DD: lengthB}. +Notes & This equation is based on the assumption that the surface and the base of a slice are straight lines (\hyperref[assumpSBSBISL]{A: Surface-Base-Slice-between-Interslice-Straight-Lines}). The soil dry unit weight ${γ_{\text{dry}}}$ and the soil saturated unit weight ${γ_{\text{Sat}}}$ are not indexed by $i$ because the soil is assumed to be homogeneous, with constant soil properties throughout (\hyperref[assumpSLH]{A: Soil-Layer-Homogeneous}). $\mathbf{b}$ is defined in \hyperref[DD:lengthB]{DD: lengthB}. \\ \midrule \\ Source & \cite{fredlund1977} \\ \midrule \\ @@ -1065,27 +1065,27 @@ \subsubsection{General Definitions} \label{GD:sliceWghtDeriv} For the case where the water table is above the slope surface, the weights come from the weight of the saturated soil. Substituting values for saturated soil into the equation for weight from \hyperref[GD:weight]{GD: weight} yields: \begin{displaymath} -{\mathbf{W}}_{i}={\mathbf{V}_{sat,i}} {γ_{Sat}} +{\mathbf{W}}_{i}={\mathbf{V}_{\text{sat},i}} {γ_{\text{Sat}}} \end{displaymath} Due to \hyperref[assumpPSC]{A: Plane-Strain-Conditions}, only two dimensions are considered, so the areas of saturated soil are considered instead of the volumes of saturated soil. Any given slice has a trapezoidal shape. The area of a trapezoid is the average of the lengths of the parallel sides multiplied by the length between the parallel sides. The parallel sides in this case are the interslice edges and the length between them is the width of the slice. Thus, the weights are defined as: \begin{displaymath} -{\mathbf{W}}_{i}={\mathbf{b}}_{i} \frac{1}{2} \left({\mathbf{y}_{slope,i}}-{\mathbf{y}_{slip,i}}+{\mathbf{y}_{slope,i-1}}-{\mathbf{y}_{slip,i-1}}\right) {γ_{Sat}} +{\mathbf{W}}_{i}={\mathbf{b}}_{i} \frac{1}{2} \left({\mathbf{y}_{\text{slope},i}}-{\mathbf{y}_{\text{slip},i}}+{\mathbf{y}_{\text{slope},i-1}}-{\mathbf{y}_{\text{slip},i-1}}\right) {γ_{\text{Sat}}} \end{displaymath} For the case where the water table is below the slip surface, the weights come from the weight of the dry soil. Substituting values for dry soil into the equation for weight from \hyperref[GD:weight]{GD: weight} yields: \begin{displaymath} -{\mathbf{W}}_{i}={\mathbf{V}_{dry,i}} {γ_{dry}} +{\mathbf{W}}_{i}={\mathbf{V}_{\text{dry},i}} {γ_{\text{dry}}} \end{displaymath} \hyperref[assumpPSC]{A: Plane-Strain-Conditions} again allows for two-dimensional analysis so the areas of dry soil are considered instead of the volumes of dry soil. The trapezoidal slice shape is the same as in the previous case, so the weights are defined as: \begin{displaymath} -{\mathbf{W}}_{i}={\mathbf{b}}_{i} \frac{1}{2} \left({\mathbf{y}_{slope,i}}-{\mathbf{y}_{slip,i}}+{\mathbf{y}_{slope,i-1}}-{\mathbf{y}_{slip,i-1}}\right) {γ_{dry}} +{\mathbf{W}}_{i}={\mathbf{b}}_{i} \frac{1}{2} \left({\mathbf{y}_{\text{slope},i}}-{\mathbf{y}_{\text{slip},i}}+{\mathbf{y}_{\text{slope},i-1}}-{\mathbf{y}_{\text{slip},i-1}}\right) {γ_{\text{dry}}} \end{displaymath} For the case where the water table is between the slope surface and slip surface, the weights are the sums of the the weights of the dry portions and weights of the saturated portions of the soil. Substituting values for dry and saturated soil into the equation for weight from \hyperref[GD:weight]{GD: weight} and adding them together yields: \begin{displaymath} -{\mathbf{W}}_{i}={\mathbf{V}_{dry,i}} {γ_{dry}}+{\mathbf{V}_{sat,i}} {γ_{Sat}} +{\mathbf{W}}_{i}={\mathbf{V}_{\text{dry},i}} {γ_{\text{dry}}}+{\mathbf{V}_{\text{sat},i}} {γ_{\text{Sat}}} \end{displaymath} \hyperref[assumpPSC]{A: Plane-Strain-Conditions} again allows for two-dimensional analysis so the areas of dry soil and areas of saturated soil are considered instead of the volumes of dry soil and volumes of saturated soil. The water table is assumed to only intersect a slice surface or base at a slice edge (\hyperref[assumpWISE]{A: Water-Intersects-Surface-Edge}, \hyperref[assumpWIBE]{A: Water-Intersects-Base-Edge}), so the dry and saturated portions each have trapezoidal shape. For the dry portion, the parallel sides of the trapezoid are the lengths between the slope surface and water table at the slice edges. For the saturated portion, the parallel sides of the trapezoid are the lengths between the water table and slip surface at the slice edges. Thus, the weights are defined as: \begin{displaymath} -{\mathbf{W}}_{i}={\mathbf{b}}_{i} \frac{1}{2} \left(\left({\mathbf{y}_{slope,i}}-{\mathbf{y}_{wt,i}}+{\mathbf{y}_{slope,i-1}}-{\mathbf{y}_{wt,i-1}}\right) {γ_{dry}}+\left({\mathbf{y}_{wt,i}}-{\mathbf{y}_{slip,i}}+{\mathbf{y}_{wt,i-1}}-{\mathbf{y}_{slip,i-1}}\right) {γ_{Sat}}\right) +{\mathbf{W}}_{i}={\mathbf{b}}_{i} \frac{1}{2} \left(\left({\mathbf{y}_{\text{slope},i}}-{\mathbf{y}_{\text{wt},i}}+{\mathbf{y}_{\text{slope},i-1}}-{\mathbf{y}_{\text{wt},i-1}}\right) {γ_{\text{dry}}}+\left({\mathbf{y}_{\text{wt},i}}-{\mathbf{y}_{\text{slip},i}}+{\mathbf{y}_{\text{wt},i-1}}-{\mathbf{y}_{\text{slip},i-1}}\right) {γ_{\text{Sat}}}\right) \end{displaymath} \par~ @@ -1131,8 +1131,8 @@ \subsubsection{General Definitions} \\ \midrule \\ Equation & \begin{displaymath} {\mathbf{U}_{b,i}}={\mathbf{ℓ}_{b,i}} {γ_{w}} \frac{1}{2} \begin{cases} - {\mathbf{y}_{wt,i}}-{\mathbf{y}_{slip,i}}+{\mathbf{y}_{wt,i-1}}-{\mathbf{y}_{slip,i-1}}, & {\mathbf{y}_{wt,i}}>{\mathbf{y}_{slip,i}}\lor{}{\mathbf{y}_{wt,i-1}}>{\mathbf{y}_{slip,i-1}}\\ -0, & {\mathbf{y}_{wt,i}}\leq{}{\mathbf{y}_{slip,i}}\land{}{\mathbf{y}_{wt,i-1}}\leq{}{\mathbf{y}_{slip,i-1}} + {\mathbf{y}_{\text{wt},i}}-{\mathbf{y}_{\text{slip},i}}+{\mathbf{y}_{\text{wt},i-1}}-{\mathbf{y}_{\text{slip},i-1}}, & {\mathbf{y}_{\text{wt},i}}>{\mathbf{y}_{\text{slip},i}}\lor{}{\mathbf{y}_{\text{wt},i-1}}>{\mathbf{y}_{\text{slip},i-1}}\\ +0, & {\mathbf{y}_{\text{wt},i}}\leq{}{\mathbf{y}_{\text{slip},i}}\land{}{\mathbf{y}_{\text{wt},i-1}}\leq{}{\mathbf{y}_{\text{slip},i-1}} \end{cases} \end{displaymath} \\ \midrule \\ @@ -1141,8 +1141,8 @@ \subsubsection{General Definitions} \item{$i$ is the index (Unitless)} \item{${\mathbf{ℓ}_{b}}$ is the total base lengths of slices (m)} \item{${γ_{w}}$ is the unit weight of water ($\frac{\text{N}}{\text{m}^{3}}$)} - \item{${\mathbf{y}_{wt}}$ is the y-coordinates of the water table (m)} - \item{${\mathbf{y}_{slip}}$ is the y-coordinates of the slip surface (m)} + \item{${\mathbf{y}_{\text{wt}}}$ is the y-coordinates of the water table (m)} + \item{${\mathbf{y}_{\text{slip}}}$ is the y-coordinates of the slip surface (m)} \end{symbDescription} \\ \midrule \\ Notes & This equation is based on the assumption that the base of a slice is a straight line (\hyperref[assumpSBSBISL]{A: Surface-Base-Slice-between-Interslice-Straight-Lines}). ${\mathbf{ℓ}_{b}}$ is defined in \hyperref[DD:lengthLb]{DD: lengthLb}. @@ -1161,11 +1161,11 @@ \subsubsection{General Definitions} \end{displaymath} The specific weight in this case is the unit weight of water ${γ_{w}}$. The height in this case is the height from the slice base to the water table. This height is measured from the midpoint of the slice because the resultant hydrostatic force is assumed to act at the slice midpoint (\hyperref[assumpHFSM]{A: Hydrostatic-Force-Slice-Midpoint}). The height at the midpoint is the average of the height at slice interface $i$ and the height at slice interface $i-1$: \begin{displaymath} -\frac{1}{2} \left({\mathbf{y}_{wt,i}}-{\mathbf{y}_{slip,i}}+{\mathbf{y}_{wt,i-1}}-{\mathbf{y}_{slip,i-1}}\right) +\frac{1}{2} \left({\mathbf{y}_{\text{wt},i}}-{\mathbf{y}_{\text{slip},i}}+{\mathbf{y}_{\text{wt},i-1}}-{\mathbf{y}_{\text{slip},i-1}}\right) \end{displaymath} Due to \hyperref[assumpPSC]{A: Plane-Strain-Conditions}, only two dimensions are considered, so the base hydrostatic forces are expressed as forces per meter. The pressures acting on the slices can thus be converted to base hydrostatic forces by multiplying by the corresponding length of the slice base ${\mathbf{ℓ}_{b,i}}$, assuming the water table does not intersect a slice base except at a slice edge (\hyperref[assumpWIBE]{A: Water-Intersects-Base-Edge}). Thus, in the case where the height of the water table is above the height of the slip surface, the base hydrostatic forces are defined as: \begin{displaymath} -{\mathbf{U}_{b,i}}={\mathbf{ℓ}_{b,i}} {γ_{w}} \frac{1}{2} \left({\mathbf{y}_{wt,i}}-{\mathbf{y}_{slip,i}}+{\mathbf{y}_{wt,i-1}}-{\mathbf{y}_{slip,i-1}}\right) +{\mathbf{U}_{b,i}}={\mathbf{ℓ}_{b,i}} {γ_{w}} \frac{1}{2} \left({\mathbf{y}_{\text{wt},i}}-{\mathbf{y}_{\text{slip},i}}+{\mathbf{y}_{\text{wt},i-1}}-{\mathbf{y}_{\text{slip},i-1}}\right) \end{displaymath} This equation is the non-zero case of \hyperref[GD:baseWtrF]{GD: baseWtrF}. The zero case is when the height of the water table is below the height of the slip surface, so there is no hydrostatic force. \par~ @@ -1182,8 +1182,8 @@ \subsubsection{General Definitions} \\ \midrule \\ Equation & \begin{displaymath} {\mathbf{U}_{t,i}}={\mathbf{ℓ}_{s,i}} {γ_{w}} \frac{1}{2} \begin{cases} - {\mathbf{y}_{wt,i}}-{\mathbf{y}_{slope,i}}+{\mathbf{y}_{wt,i-1}}-{\mathbf{y}_{slope,i-1}}, & {\mathbf{y}_{wt,i}}>{\mathbf{y}_{slope,i}}\lor{}{\mathbf{y}_{wt,i-1}}>{\mathbf{y}_{slope,i-1}}\\ -0, & {\mathbf{y}_{wt,i}}\leq{}{\mathbf{y}_{slope,i}}\land{}{\mathbf{y}_{wt,i-1}}\leq{}{\mathbf{y}_{slope,i-1}} + {\mathbf{y}_{\text{wt},i}}-{\mathbf{y}_{\text{slope},i}}+{\mathbf{y}_{\text{wt},i-1}}-{\mathbf{y}_{\text{slope},i-1}}, & {\mathbf{y}_{\text{wt},i}}>{\mathbf{y}_{\text{slope},i}}\lor{}{\mathbf{y}_{\text{wt},i-1}}>{\mathbf{y}_{\text{slope},i-1}}\\ +0, & {\mathbf{y}_{\text{wt},i}}\leq{}{\mathbf{y}_{\text{slope},i}}\land{}{\mathbf{y}_{\text{wt},i-1}}\leq{}{\mathbf{y}_{\text{slope},i-1}} \end{cases} \end{displaymath} \\ \midrule \\ @@ -1192,8 +1192,8 @@ \subsubsection{General Definitions} \item{$i$ is the index (Unitless)} \item{${\mathbf{ℓ}_{s}}$ is the surface lengths of slices (m)} \item{${γ_{w}}$ is the unit weight of water ($\frac{\text{N}}{\text{m}^{3}}$)} - \item{${\mathbf{y}_{wt}}$ is the y-coordinates of the water table (m)} - \item{${\mathbf{y}_{slope}}$ is the y-coordinates of the slope (m)} + \item{${\mathbf{y}_{\text{wt}}}$ is the y-coordinates of the water table (m)} + \item{${\mathbf{y}_{\text{slope}}}$ is the y-coordinates of the slope (m)} \end{symbDescription} \\ \midrule \\ Notes & This equation is based on the assumption that the surface of a slice is a straight line (\hyperref[assumpSBSBISL]{A: Surface-Base-Slice-between-Interslice-Straight-Lines}). ${\mathbf{ℓ}_{s}}$ is defined in \hyperref[DD:lengthLs]{DD: lengthLs}. @@ -1212,11 +1212,11 @@ \subsubsection{General Definitions} \end{displaymath} The specific weight in this case is the unit weight of water ${γ_{w}}$. The height in this case is the height from the slice surface to the water table. This height is measured from the midpoint of the slice because the resultant hydrostatic force is assumed to act at the slice midpoint (\hyperref[assumpHFSM]{A: Hydrostatic-Force-Slice-Midpoint}). The height at the midpoint is the average of the height at slice interface $i$ and the height at slice interface $i-1$: \begin{displaymath} -\frac{1}{2} \left({\mathbf{y}_{wt,i}}-{\mathbf{y}_{slope,i}}+{\mathbf{y}_{wt,i-1}}-{\mathbf{y}_{slope,i-1}}\right) +\frac{1}{2} \left({\mathbf{y}_{\text{wt},i}}-{\mathbf{y}_{\text{slope},i}}+{\mathbf{y}_{\text{wt},i-1}}-{\mathbf{y}_{\text{slope},i-1}}\right) \end{displaymath} Due to \hyperref[assumpPSC]{A: Plane-Strain-Conditions}, only two dimensions are considered, so the surface hydrostatic forces are expressed as forces per meter. The pressures acting on the slices can thus be converted to surface hydrostatic forces by multiplying by the corresponding length of the slice surface ${\mathbf{ℓ}_{s,i}}$, assuming the water table does not intersect a slice surface except at a slice edge (\hyperref[assumpWISE]{A: Water-Intersects-Surface-Edge}). Thus, in the case where the height of the water table is above the height of the slope surface, the surface hydrostatic forces are defined as: \begin{displaymath} -{\mathbf{U}_{t,i}}={\mathbf{ℓ}_{s,i}} {γ_{w}} \frac{1}{2} \left({\mathbf{y}_{wt,i}}-{\mathbf{y}_{slope,i}}+{\mathbf{y}_{wt,i-1}}-{\mathbf{y}_{slope,i-1}}\right) +{\mathbf{U}_{t,i}}={\mathbf{ℓ}_{s,i}} {γ_{w}} \frac{1}{2} \left({\mathbf{y}_{\text{wt},i}}-{\mathbf{y}_{\text{slope},i}}+{\mathbf{y}_{\text{wt},i-1}}-{\mathbf{y}_{\text{slope},i-1}}\right) \end{displaymath} This equation is the non-zero case of \hyperref[GD:srfWtrF]{GD: srfWtrF}. The zero case is when the height of the water table is below the height of the slope surface, so there is no hydrostatic force. \subsubsection{Data Definitions} @@ -1238,19 +1238,19 @@ \subsubsection{Data Definitions} \\ \midrule \\ Equation & \begin{displaymath} \mathbf{H}=\begin{cases} - \frac{\left({\mathbf{y}_{slope,i}}-{\mathbf{y}_{slip,i}}\right)^{2}}{2} {γ_{w}}+\left({\mathbf{y}_{wt,i}}-{\mathbf{y}_{slope,i}}\right)^{2} {γ_{w}}, & {\mathbf{y}_{wt,i}}\geq{}{\mathbf{y}_{slope,i}}\\ -\frac{\left({\mathbf{y}_{wt,i}}-{\mathbf{y}_{slip,i}}\right)^{2}}{2} {γ_{w}}, & {\mathbf{y}_{slope,i}}>{\mathbf{y}_{wt,i}}>{\mathbf{y}_{slip,i}}\\ -0, & {\mathbf{y}_{wt,i}}\leq{}{\mathbf{y}_{slip,i}} + \frac{\left({\mathbf{y}_{\text{slope},i}}-{\mathbf{y}_{\text{slip},i}}\right)^{2}}{2} {γ_{w}}+\left({\mathbf{y}_{\text{wt},i}}-{\mathbf{y}_{\text{slope},i}}\right)^{2} {γ_{w}}, & {\mathbf{y}_{\text{wt},i}}\geq{}{\mathbf{y}_{\text{slope},i}}\\ +\frac{\left({\mathbf{y}_{\text{wt},i}}-{\mathbf{y}_{\text{slip},i}}\right)^{2}}{2} {γ_{w}}, & {\mathbf{y}_{\text{slope},i}}>{\mathbf{y}_{\text{wt},i}}>{\mathbf{y}_{\text{slip},i}}\\ +0, & {\mathbf{y}_{\text{wt},i}}\leq{}{\mathbf{y}_{\text{slip},i}} \end{cases} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} \item{$\mathbf{H}$ is the interslice normal water forces ($\frac{\text{N}}{\text{m}}$)} - \item{${\mathbf{y}_{slope}}$ is the y-coordinates of the slope (m)} + \item{${\mathbf{y}_{\text{slope}}}$ is the y-coordinates of the slope (m)} \item{$i$ is the index (Unitless)} - \item{${\mathbf{y}_{slip}}$ is the y-coordinates of the slip surface (m)} + \item{${\mathbf{y}_{\text{slip}}}$ is the y-coordinates of the slip surface (m)} \item{${γ_{w}}$ is the unit weight of water ($\frac{\text{N}}{\text{m}^{3}}$)} - \item{${\mathbf{y}_{wt}}$ is the y-coordinates of the water table (m)} + \item{${\mathbf{y}_{\text{wt}}}$ is the y-coordinates of the water table (m)} \end{symbDescription} \\ \midrule \\ Source & \cite{fredlund1977} @@ -1274,14 +1274,14 @@ \subsubsection{Data Definitions} Units & ${}^{\circ}$ \\ \midrule \\ Equation & \begin{displaymath} - \mathbf{α}=\arctan\left(\frac{{\mathbf{y}_{slip,i}}-{\mathbf{y}_{slip,i-1}}}{{\mathbf{x}_{slip,i}}-{\mathbf{x}_{slip,i-1}}}\right) + \mathbf{α}=\arctan\left(\frac{{\mathbf{y}_{\text{slip},i}}-{\mathbf{y}_{\text{slip},i-1}}}{{\mathbf{x}_{\text{slip},i}}-{\mathbf{x}_{\text{slip},i-1}}}\right) \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} \item{$\mathbf{α}$ is the base angles (${}^{\circ}$)} - \item{${\mathbf{y}_{slip}}$ is the y-coordinates of the slip surface (m)} + \item{${\mathbf{y}_{\text{slip}}}$ is the y-coordinates of the slip surface (m)} \item{$i$ is the index (Unitless)} - \item{${\mathbf{x}_{slip}}$ is the x-coordinates of the slip surface (m)} + \item{${\mathbf{x}_{\text{slip}}}$ is the x-coordinates of the slip surface (m)} \end{symbDescription} \\ \midrule \\ Notes & This equation is based on the assumption that the base of a slice is a straight line (\hyperref[assumpSBSBISL]{A: Surface-Base-Slice-between-Interslice-Straight-Lines}). @@ -1307,14 +1307,14 @@ \subsubsection{Data Definitions} Units & ${}^{\circ}$ \\ \midrule \\ Equation & \begin{displaymath} - \mathbf{β}=\arctan\left(\frac{{\mathbf{y}_{slope,i}}-{\mathbf{y}_{slope,i-1}}}{{\mathbf{x}_{slope,i}}-{\mathbf{x}_{slope,i-1}}}\right) + \mathbf{β}=\arctan\left(\frac{{\mathbf{y}_{\text{slope},i}}-{\mathbf{y}_{\text{slope},i-1}}}{{\mathbf{x}_{\text{slope},i}}-{\mathbf{x}_{\text{slope},i-1}}}\right) \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} \item{$\mathbf{β}$ is the surface angles (${}^{\circ}$)} - \item{${\mathbf{y}_{slope}}$ is the y-coordinates of the slope (m)} + \item{${\mathbf{y}_{\text{slope}}}$ is the y-coordinates of the slope (m)} \item{$i$ is the index (Unitless)} - \item{${\mathbf{x}_{slope}}$ is the x-coordinates of the slope (m)} + \item{${\mathbf{x}_{\text{slope}}}$ is the x-coordinates of the slope (m)} \end{symbDescription} \\ \midrule \\ Notes & This equation is based on the assumption that the surface of a slice is a straight line (\hyperref[assumpSBSBISL]{A: Surface-Base-Slice-between-Interslice-Straight-Lines}). @@ -1340,12 +1340,12 @@ \subsubsection{Data Definitions} Units & m \\ \midrule \\ Equation & \begin{displaymath} - \mathbf{b}={\mathbf{x}_{slip,i}}-{\mathbf{x}_{slip,i-1}} + \mathbf{b}={\mathbf{x}_{\text{slip},i}}-{\mathbf{x}_{\text{slip},i-1}} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} \item{$\mathbf{b}$ is the base width of slices (m)} - \item{${\mathbf{x}_{slip}}$ is the x-coordinates of the slip surface (m)} + \item{${\mathbf{x}_{\text{slip}}}$ is the x-coordinates of the slip surface (m)} \item{$i$ is the index (Unitless)} \end{symbDescription} \\ \midrule \\ @@ -1533,14 +1533,14 @@ \subsubsection{Data Definitions} Equation & \begin{displaymath} \mathbf{f}=\begin{cases} 1, & const_f\\ -\sin\left(π \frac{{\mathbf{x}_{slip,i}}-{\mathbf{x}_{slip,0}}}{{\mathbf{x}_{slip,n}}-{\mathbf{x}_{slip,0}}}\right), & \neg{}const_f +\sin\left(π \frac{{\mathbf{x}_{\text{slip},i}}-{\mathbf{x}_{\text{slip},0}}}{{\mathbf{x}_{\text{slip},n}}-{\mathbf{x}_{\text{slip},0}}}\right), & \neg{}const_f \end{cases} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} \item{$\mathbf{f}$ is the interslice normal to shear force ratio variation function (Unitless)} \item{$π$ is the ratio of circumference to diameter for any circle (Unitless)} - \item{${\mathbf{x}_{slip}}$ is the x-coordinates of the slip surface (m)} + \item{${\mathbf{x}_{\text{slip}}}$ is the x-coordinates of the slip surface (m)} \item{$i$ is the index (Unitless)} \item{$n$ is the number of slices (Unitless)} \item{$const_f$ is the decision on f (Unitless)} @@ -1567,7 +1567,7 @@ \subsubsection{Data Definitions} Units & Unitless \\ \midrule \\ Equation & \begin{displaymath} - \mathbf{Φ}=\left(λ {\mathbf{f}}_{i} \cos\left({\mathbf{α}}_{i}\right)-\sin\left({\mathbf{α}}_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\mathbf{f}}_{i} \sin\left({\mathbf{α}}_{i}\right)+\cos\left({\mathbf{α}}_{i}\right)\right) {F_{S}} + \mathbf{Φ}=\left(λ {\mathbf{f}}_{i} \cos\left({\mathbf{α}}_{i}\right)-\sin\left({\mathbf{α}}_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\mathbf{f}}_{i} \sin\left({\mathbf{α}}_{i}\right)+\cos\left({\mathbf{α}}_{i}\right)\right) {F_{\text{S}}} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} @@ -1577,7 +1577,7 @@ \subsubsection{Data Definitions} \item{$i$ is the index (Unitless)} \item{$\mathbf{α}$ is the base angles (${}^{\circ}$)} \item{$φ'$ is the effective angle of friction (${}^{\circ}$)} - \item{${F_{S}}$ is the factor of safety (Unitless)} + \item{${F_{\text{S}}}$ is the factor of safety (Unitless)} \end{symbDescription} \\ \midrule \\ Notes & $\mathbf{f}$ is defined in \hyperref[DD:ratioVariation]{DD: ratioVariation} and $\mathbf{α}$ is defined in \hyperref[DD:angleA]{DD: angleA}. @@ -1603,7 +1603,7 @@ \subsubsection{Data Definitions} Units & Unitless \\ \midrule \\ Equation & \begin{displaymath} - \mathbf{Ψ}=\frac{\left(λ {\mathbf{f}}_{i} \cos\left({\mathbf{α}}_{i}\right)-\sin\left({\mathbf{α}}_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\mathbf{f}}_{i} \sin\left({\mathbf{α}}_{i}\right)+\cos\left({\mathbf{α}}_{i}\right)\right) {F_{S}}}{{\mathbf{Φ}}_{i-1}} + \mathbf{Ψ}=\frac{\left(λ {\mathbf{f}}_{i} \cos\left({\mathbf{α}}_{i}\right)-\sin\left({\mathbf{α}}_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\mathbf{f}}_{i} \sin\left({\mathbf{α}}_{i}\right)+\cos\left({\mathbf{α}}_{i}\right)\right) {F_{\text{S}}}}{{\mathbf{Φ}}_{i-1}} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} @@ -1613,7 +1613,7 @@ \subsubsection{Data Definitions} \item{$i$ is the index (Unitless)} \item{$\mathbf{α}$ is the base angles (${}^{\circ}$)} \item{$φ'$ is the effective angle of friction (${}^{\circ}$)} - \item{${F_{S}}$ is the factor of safety (Unitless)} + \item{${F_{\text{S}}}$ is the factor of safety (Unitless)} \item{$\mathbf{Φ}$ is the first function for incorporating interslice forces into shear force (Unitless)} \end{symbDescription} \\ \midrule \\ @@ -1700,14 +1700,14 @@ \subsubsection{Data Definitions} Units & m \\ \midrule \\ Equation & \begin{displaymath} - {\mathbf{h}^{R}}={\mathbf{y}_{slope,i}}-{\mathbf{y}_{slip,i}} + {\mathbf{h}^{R}}={\mathbf{y}_{\text{slope},i}}-{\mathbf{y}_{\text{slip},i}} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} \item{${\mathbf{h}^{R}}$ is the heights of the right side of slices (m)} - \item{${\mathbf{y}_{slope}}$ is the y-coordinates of the slope (m)} + \item{${\mathbf{y}_{\text{slope}}}$ is the y-coordinates of the slope (m)} \item{$i$ is the index (Unitless)} - \item{${\mathbf{y}_{slip}}$ is the y-coordinates of the slip surface (m)} + \item{${\mathbf{y}_{\text{slip}}}$ is the y-coordinates of the slip surface (m)} \end{symbDescription} \\ \midrule \\ Source & \cite{fredlund1977} @@ -1731,14 +1731,14 @@ \subsubsection{Data Definitions} Units & m \\ \midrule \\ Equation & \begin{displaymath} - {\mathbf{h}^{L}}={\mathbf{y}_{slope,i-1}}-{\mathbf{y}_{slip,i-1}} + {\mathbf{h}^{L}}={\mathbf{y}_{\text{slope},i-1}}-{\mathbf{y}_{\text{slip},i-1}} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} \item{${\mathbf{h}^{L}}$ is the heights of the left side of slices (m)} - \item{${\mathbf{y}_{slope}}$ is the y-coordinates of the slope (m)} + \item{${\mathbf{y}_{\text{slope}}}$ is the y-coordinates of the slope (m)} \item{$i$ is the index (Unitless)} - \item{${\mathbf{y}_{slip}}$ is the y-coordinates of the slip surface (m)} + \item{${\mathbf{y}_{\text{slip}}}$ is the y-coordinates of the slip surface (m)} \end{symbDescription} \\ \midrule \\ Source & \cite{fredlund1977} @@ -1762,20 +1762,20 @@ \subsubsection{Instance Models} \\ \midrule \\ Label & Factor of safety \\ \midrule \\ -Input & ${\mathbf{x}_{slope}}$, ${\mathbf{y}_{slope}}$, ${\mathbf{y}_{wt}}$, $c'$, $φ'$, ${γ_{dry}}$, ${γ_{Sat}}$, ${γ_{w}}$, ${\mathbf{x}_{slip}}$, ${\mathbf{y}_{slip}}$, $const_f$ +Input & ${\mathbf{x}_{\text{slope}}}$, ${\mathbf{y}_{\text{slope}}}$, ${\mathbf{y}_{\text{wt}}}$, $c'$, $φ'$, ${γ_{\text{dry}}}$, ${γ_{\text{Sat}}}$, ${γ_{w}}$, ${\mathbf{x}_{\text{slip}}}$, ${\mathbf{y}_{\text{slip}}}$, $const_f$ \\ \midrule \\ -Output & ${F_{S}}$ +Output & ${F_{\text{S}}}$ \\ \midrule \\ Input Constraints & \\ \midrule \\ Output Constraints & \\ \midrule \\ Equation & \begin{displaymath} - {F_{S}}=\frac{\displaystyle\sum_{i=1}^{n-1}{{\mathbf{R}}_{i} \displaystyle\prod_{v=i}^{n-1}{{\mathbf{Ψ}}_{v}}}+{\mathbf{R}}_{n}}{\displaystyle\sum_{i=1}^{n-1}{{\mathbf{T}}_{i} \displaystyle\prod_{v=i}^{n-1}{{\mathbf{Ψ}}_{v}}}+{\mathbf{T}}_{n}} + {F_{\text{S}}}=\frac{\displaystyle\sum_{i=1}^{n-1}{{\mathbf{R}}_{i} \displaystyle\prod_{v=i}^{n-1}{{\mathbf{Ψ}}_{v}}}+{\mathbf{R}}_{n}}{\displaystyle\sum_{i=1}^{n-1}{{\mathbf{T}}_{i} \displaystyle\prod_{v=i}^{n-1}{{\mathbf{Ψ}}_{v}}}+{\mathbf{T}}_{n}} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} - \item{${F_{S}}$ is the factor of safety (Unitless)} + \item{${F_{\text{S}}}$ is the factor of safety (Unitless)} \item{$\mathbf{R}$ is the resistive shear forces without the influence of interslice forces ($\frac{\text{N}}{\text{m}}$)} \item{$i$ is the index (Unitless)} \item{$\mathbf{Ψ}$ is the second function for incorporating interslice forces into shear force (Unitless)} @@ -1796,7 +1796,7 @@ \subsubsection{Instance Models} \label{IM:fctSftyDeriv} The mobilized shear force defined in \hyperref[GD:bsShrFEq]{GD: bsShrFEq} can be substituted into the definition of mobilized shear force based on the factor of safety, from \hyperref[GD:mobShr]{GD: mobShr} yielding Equation (1) below: \begin{displaymath} -\left({\mathbf{W}}_{i}-{\mathbf{X}}_{i-1}+{\mathbf{X}}_{i}+{\mathbf{U}_{t,i}} \cos\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \cos\left({\mathbf{ω}}_{i}\right)\right) \sin\left({\mathbf{α}}_{i}\right)-\left(-{K_{c}} {\mathbf{W}}_{i}-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}-{\mathbf{H}}_{i}+{\mathbf{H}}_{i-1}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \sin\left({\mathbf{ω}}_{i}\right)\right) \cos\left({\mathbf{α}}_{i}\right)=\frac{{\mathbf{N'}}_{i} \tan\left(φ'\right)+c' {\mathbf{ℓ}_{b,i}}}{{F_{S}}} +\left({\mathbf{W}}_{i}-{\mathbf{X}}_{i-1}+{\mathbf{X}}_{i}+{\mathbf{U}_{t,i}} \cos\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \cos\left({\mathbf{ω}}_{i}\right)\right) \sin\left({\mathbf{α}}_{i}\right)-\left(-{K_{c}} {\mathbf{W}}_{i}-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}-{\mathbf{H}}_{i}+{\mathbf{H}}_{i-1}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \sin\left({\mathbf{ω}}_{i}\right)\right) \cos\left({\mathbf{α}}_{i}\right)=\frac{{\mathbf{N'}}_{i} \tan\left(φ'\right)+c' {\mathbf{ℓ}_{b,i}}}{{F_{\text{S}}}} \end{displaymath} An expression for the effective normal forces, $\mathbf{N'}$, can be derived by substituting the normal forces equilibrium from \hyperref[GD:normForcEq]{GD: normForcEq} into the definition for effective normal forces from \hyperref[GD:resShearWO]{GD: resShearWO}. This results in Equation (2): \begin{displaymath} @@ -1804,89 +1804,89 @@ \subsubsection{Instance Models} \end{displaymath} Substituting Equation (2) into Equation (1) gives: \begin{displaymath} -\left({\mathbf{W}}_{i}-{\mathbf{X}}_{i-1}+{\mathbf{X}}_{i}+{\mathbf{U}_{t,i}} \cos\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \cos\left({\mathbf{ω}}_{i}\right)\right) \sin\left({\mathbf{α}}_{i}\right)-\left(-{K_{c}} {\mathbf{W}}_{i}-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}-{\mathbf{H}}_{i}+{\mathbf{H}}_{i-1}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \sin\left({\mathbf{ω}}_{i}\right)\right) \cos\left({\mathbf{α}}_{i}\right)=\frac{\left(\left({\mathbf{W}}_{i}-{\mathbf{X}}_{i-1}+{\mathbf{X}}_{i}+{\mathbf{U}_{t,i}} \cos\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \cos\left({\mathbf{ω}}_{i}\right)\right) \cos\left({\mathbf{α}}_{i}\right)+\left(-{K_{c}} {\mathbf{W}}_{i}-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}-{\mathbf{H}}_{i}+{\mathbf{H}}_{i-1}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \sin\left({\mathbf{ω}}_{i}\right)\right) \sin\left({\mathbf{α}}_{i}\right)-{\mathbf{U}_{b,i}}\right) \tan\left(φ'\right)+c' {\mathbf{ℓ}_{b,i}}}{{F_{S}}} +\left({\mathbf{W}}_{i}-{\mathbf{X}}_{i-1}+{\mathbf{X}}_{i}+{\mathbf{U}_{t,i}} \cos\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \cos\left({\mathbf{ω}}_{i}\right)\right) \sin\left({\mathbf{α}}_{i}\right)-\left(-{K_{c}} {\mathbf{W}}_{i}-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}-{\mathbf{H}}_{i}+{\mathbf{H}}_{i-1}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \sin\left({\mathbf{ω}}_{i}\right)\right) \cos\left({\mathbf{α}}_{i}\right)=\frac{\left(\left({\mathbf{W}}_{i}-{\mathbf{X}}_{i-1}+{\mathbf{X}}_{i}+{\mathbf{U}_{t,i}} \cos\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \cos\left({\mathbf{ω}}_{i}\right)\right) \cos\left({\mathbf{α}}_{i}\right)+\left(-{K_{c}} {\mathbf{W}}_{i}-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}-{\mathbf{H}}_{i}+{\mathbf{H}}_{i-1}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \sin\left({\mathbf{ω}}_{i}\right)\right) \sin\left({\mathbf{α}}_{i}\right)-{\mathbf{U}_{b,i}}\right) \tan\left(φ'\right)+c' {\mathbf{ℓ}_{b,i}}}{{F_{\text{S}}}} \end{displaymath} Since the interslice shear forces $\mathbf{X}$ and interslice normal forces $\mathbf{G}$ are unknown, they are separated from the other terms as follows: \begin{displaymath} -\left({\mathbf{W}}_{i}+{\mathbf{U}_{t,i}} \cos\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \cos\left({\mathbf{ω}}_{i}\right)\right) \sin\left({\mathbf{α}}_{i}\right)-\left(-{K_{c}} {\mathbf{W}}_{i}-{\mathbf{H}}_{i}+{\mathbf{H}}_{i-1}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \sin\left({\mathbf{ω}}_{i}\right)\right) \cos\left({\mathbf{α}}_{i}\right)-\left(-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}\right) \cos\left({\mathbf{α}}_{i}\right)+\left(-{\mathbf{X}}_{i-1}+{\mathbf{X}}_{i}\right) \sin\left({\mathbf{α}}_{i}\right)=\frac{\left(\left({\mathbf{W}}_{i}+{\mathbf{U}_{t,i}} \cos\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \cos\left({\mathbf{ω}}_{i}\right)\right) \cos\left({\mathbf{α}}_{i}\right)+\left(-{K_{c}} {\mathbf{W}}_{i}-{\mathbf{H}}_{i}+{\mathbf{H}}_{i-1}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \sin\left({\mathbf{ω}}_{i}\right)\right) \sin\left({\mathbf{α}}_{i}\right)+\left(-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}\right) \sin\left({\mathbf{α}}_{i}\right)+\left(-{\mathbf{X}}_{i-1}+{\mathbf{X}}_{i}\right) \cos\left({\mathbf{α}}_{i}\right)-{\mathbf{U}_{b,i}}\right) \tan\left(φ'\right)+c' {\mathbf{ℓ}_{b,i}}}{{F_{S}}} +\left({\mathbf{W}}_{i}+{\mathbf{U}_{t,i}} \cos\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \cos\left({\mathbf{ω}}_{i}\right)\right) \sin\left({\mathbf{α}}_{i}\right)-\left(-{K_{c}} {\mathbf{W}}_{i}-{\mathbf{H}}_{i}+{\mathbf{H}}_{i-1}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \sin\left({\mathbf{ω}}_{i}\right)\right) \cos\left({\mathbf{α}}_{i}\right)-\left(-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}\right) \cos\left({\mathbf{α}}_{i}\right)+\left(-{\mathbf{X}}_{i-1}+{\mathbf{X}}_{i}\right) \sin\left({\mathbf{α}}_{i}\right)=\frac{\left(\left({\mathbf{W}}_{i}+{\mathbf{U}_{t,i}} \cos\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \cos\left({\mathbf{ω}}_{i}\right)\right) \cos\left({\mathbf{α}}_{i}\right)+\left(-{K_{c}} {\mathbf{W}}_{i}-{\mathbf{H}}_{i}+{\mathbf{H}}_{i-1}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right)+{\mathbf{Q}}_{i} \sin\left({\mathbf{ω}}_{i}\right)\right) \sin\left({\mathbf{α}}_{i}\right)+\left(-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}\right) \sin\left({\mathbf{α}}_{i}\right)+\left(-{\mathbf{X}}_{i-1}+{\mathbf{X}}_{i}\right) \cos\left({\mathbf{α}}_{i}\right)-{\mathbf{U}_{b,i}}\right) \tan\left(φ'\right)+c' {\mathbf{ℓ}_{b,i}}}{{F_{\text{S}}}} \end{displaymath} Applying assumptions \hyperref[assumpSF]{A: Seismic-Force} and \hyperref[assumpSL]{A: Surface-Load}, which state that the seismic coefficient and the external forces, respectively, are zero, allows for further simplification as shown below: \begin{displaymath} -\left({\mathbf{W}}_{i}+{\mathbf{U}_{t,i}} \cos\left({\mathbf{β}}_{i}\right)\right) \sin\left({\mathbf{α}}_{i}\right)-\left(-{\mathbf{H}}_{i}+{\mathbf{H}}_{i-1}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right)\right) \cos\left({\mathbf{α}}_{i}\right)-\left(-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}\right) \cos\left({\mathbf{α}}_{i}\right)+\left(-{\mathbf{X}}_{i-1}+{\mathbf{X}}_{i}\right) \sin\left({\mathbf{α}}_{i}\right)=\frac{\left(\left({\mathbf{W}}_{i}+{\mathbf{U}_{t,i}} \cos\left({\mathbf{β}}_{i}\right)\right) \cos\left({\mathbf{α}}_{i}\right)+\left(-{\mathbf{H}}_{i}+{\mathbf{H}}_{i-1}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right)\right) \sin\left({\mathbf{α}}_{i}\right)+\left(-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}\right) \sin\left({\mathbf{α}}_{i}\right)+\left(-{\mathbf{X}}_{i-1}+{\mathbf{X}}_{i}\right) \cos\left({\mathbf{α}}_{i}\right)-{\mathbf{U}_{b,i}}\right) \tan\left(φ'\right)+c' {\mathbf{ℓ}_{b,i}}}{{F_{S}}} +\left({\mathbf{W}}_{i}+{\mathbf{U}_{t,i}} \cos\left({\mathbf{β}}_{i}\right)\right) \sin\left({\mathbf{α}}_{i}\right)-\left(-{\mathbf{H}}_{i}+{\mathbf{H}}_{i-1}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right)\right) \cos\left({\mathbf{α}}_{i}\right)-\left(-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}\right) \cos\left({\mathbf{α}}_{i}\right)+\left(-{\mathbf{X}}_{i-1}+{\mathbf{X}}_{i}\right) \sin\left({\mathbf{α}}_{i}\right)=\frac{\left(\left({\mathbf{W}}_{i}+{\mathbf{U}_{t,i}} \cos\left({\mathbf{β}}_{i}\right)\right) \cos\left({\mathbf{α}}_{i}\right)+\left(-{\mathbf{H}}_{i}+{\mathbf{H}}_{i-1}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right)\right) \sin\left({\mathbf{α}}_{i}\right)+\left(-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}\right) \sin\left({\mathbf{α}}_{i}\right)+\left(-{\mathbf{X}}_{i-1}+{\mathbf{X}}_{i}\right) \cos\left({\mathbf{α}}_{i}\right)-{\mathbf{U}_{b,i}}\right) \tan\left(φ'\right)+c' {\mathbf{ℓ}_{b,i}}}{{F_{\text{S}}}} \end{displaymath} The definitions of \hyperref[GD:resShearWO]{GD: resShearWO} and \hyperref[GD:mobShearWO]{GD: mobShearWO} are present in this equation, and thus can be replaced by ${\mathbf{R}}_{i}$ and ${\mathbf{T}}_{i}$, respectively: \begin{displaymath} -{\mathbf{T}}_{i}+\left(-{\mathbf{X}}_{i-1}+{\mathbf{X}}_{i}\right) \sin\left({\mathbf{α}}_{i}\right)-\left(-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}\right) \cos\left({\mathbf{α}}_{i}\right)=\frac{{\mathbf{R}}_{i}+\left(\left(-{\mathbf{X}}_{i-1}+{\mathbf{X}}_{i}\right) \cos\left({\mathbf{α}}_{i}\right)+\left(-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}\right) \sin\left({\mathbf{α}}_{i}\right)\right) \tan\left(φ'\right)}{{F_{S}}} +{\mathbf{T}}_{i}+\left(-{\mathbf{X}}_{i-1}+{\mathbf{X}}_{i}\right) \sin\left({\mathbf{α}}_{i}\right)-\left(-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}\right) \cos\left({\mathbf{α}}_{i}\right)=\frac{{\mathbf{R}}_{i}+\left(\left(-{\mathbf{X}}_{i-1}+{\mathbf{X}}_{i}\right) \cos\left({\mathbf{α}}_{i}\right)+\left(-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}\right) \sin\left({\mathbf{α}}_{i}\right)\right) \tan\left(φ'\right)}{{F_{\text{S}}}} \end{displaymath} The interslice shear forces $\mathbf{X}$ can be expressed in terms of the interslice normal forces $\mathbf{G}$ using \hyperref[assumpINSFL]{A: Interslice-Norm-Shear-Forces-Linear} and \hyperref[GD:normShrR]{GD: normShrR}, resulting in: \begin{displaymath} -{\mathbf{T}}_{i}+\left(-λ {\mathbf{f}}_{i-1} {\mathbf{G}}_{i-1}+λ {\mathbf{f}}_{i} {\mathbf{G}}_{i}\right) \sin\left({\mathbf{α}}_{i}\right)-\left(-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}\right) \cos\left({\mathbf{α}}_{i}\right)=\frac{{\mathbf{R}}_{i}+\left(\left(-λ {\mathbf{f}}_{i-1} {\mathbf{G}}_{i-1}+λ {\mathbf{f}}_{i} {\mathbf{G}}_{i}\right) \cos\left({\mathbf{α}}_{i}\right)+\left(-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}\right) \sin\left({\mathbf{α}}_{i}\right)\right) \tan\left(φ'\right)}{{F_{S}}} +{\mathbf{T}}_{i}+\left(-λ {\mathbf{f}}_{i-1} {\mathbf{G}}_{i-1}+λ {\mathbf{f}}_{i} {\mathbf{G}}_{i}\right) \sin\left({\mathbf{α}}_{i}\right)-\left(-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}\right) \cos\left({\mathbf{α}}_{i}\right)=\frac{{\mathbf{R}}_{i}+\left(\left(-λ {\mathbf{f}}_{i-1} {\mathbf{G}}_{i-1}+λ {\mathbf{f}}_{i} {\mathbf{G}}_{i}\right) \cos\left({\mathbf{α}}_{i}\right)+\left(-{\mathbf{G}}_{i}+{\mathbf{G}}_{i-1}\right) \sin\left({\mathbf{α}}_{i}\right)\right) \tan\left(φ'\right)}{{F_{\text{S}}}} \end{displaymath} Rearranging yields the following: \begin{displaymath} -{\mathbf{G}}_{i} \left(\left(λ {\mathbf{f}}_{i} \cos\left({\mathbf{α}}_{i}\right)-\sin\left({\mathbf{α}}_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\mathbf{f}}_{i} \sin\left({\mathbf{α}}_{i}\right)+\cos\left({\mathbf{α}}_{i}\right)\right) {F_{S}}\right)={\mathbf{G}}_{i-1} \left(\left(λ {\mathbf{f}}_{i-1} \cos\left({\mathbf{α}}_{i}\right)-\sin\left({\mathbf{α}}_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\mathbf{f}}_{i-1} \sin\left({\mathbf{α}}_{i}\right)+\cos\left({\mathbf{α}}_{i}\right)\right) {F_{S}}\right)+{F_{S}} {\mathbf{T}}_{i}-{\mathbf{R}}_{i} +{\mathbf{G}}_{i} \left(\left(λ {\mathbf{f}}_{i} \cos\left({\mathbf{α}}_{i}\right)-\sin\left({\mathbf{α}}_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\mathbf{f}}_{i} \sin\left({\mathbf{α}}_{i}\right)+\cos\left({\mathbf{α}}_{i}\right)\right) {F_{\text{S}}}\right)={\mathbf{G}}_{i-1} \left(\left(λ {\mathbf{f}}_{i-1} \cos\left({\mathbf{α}}_{i}\right)-\sin\left({\mathbf{α}}_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\mathbf{f}}_{i-1} \sin\left({\mathbf{α}}_{i}\right)+\cos\left({\mathbf{α}}_{i}\right)\right) {F_{\text{S}}}\right)+{F_{\text{S}}} {\mathbf{T}}_{i}-{\mathbf{R}}_{i} \end{displaymath} The definitions for $\mathbf{Φ}$ and $\mathbf{Ψ}$ from \hyperref[DD:convertFunc1]{DD: convertFunc1} and \hyperref[DD:convertFunc2]{DD: convertFunc2} simplify the above to Equation (3): \begin{displaymath} -{\mathbf{G}}_{i} {\mathbf{Φ}}_{i}={\mathbf{Ψ}}_{i-1} {\mathbf{G}}_{i-1} {\mathbf{Φ}}_{i-1}+{F_{S}} {\mathbf{T}}_{i}-{\mathbf{R}}_{i} +{\mathbf{G}}_{i} {\mathbf{Φ}}_{i}={\mathbf{Ψ}}_{i-1} {\mathbf{G}}_{i-1} {\mathbf{Φ}}_{i-1}+{F_{\text{S}}} {\mathbf{T}}_{i}-{\mathbf{R}}_{i} \end{displaymath} Versions of Equation (3) instantiated for slices 1 to $n$ are shown below: \begin{displaymath} -{\mathbf{G}}_{1} {\mathbf{Φ}}_{1}={\mathbf{Ψ}}_{0} {\mathbf{G}}_{0} {\mathbf{Φ}}_{0}+{F_{S}} {\mathbf{T}}_{1}-{\mathbf{R}}_{1} +{\mathbf{G}}_{1} {\mathbf{Φ}}_{1}={\mathbf{Ψ}}_{0} {\mathbf{G}}_{0} {\mathbf{Φ}}_{0}+{F_{\text{S}}} {\mathbf{T}}_{1}-{\mathbf{R}}_{1} \end{displaymath} \begin{displaymath} -{\mathbf{G}}_{2} {\mathbf{Φ}}_{2}={\mathbf{Ψ}}_{1} {\mathbf{G}}_{1} {\mathbf{Φ}}_{1}+{F_{S}} {\mathbf{T}}_{2}-{\mathbf{R}}_{2} +{\mathbf{G}}_{2} {\mathbf{Φ}}_{2}={\mathbf{Ψ}}_{1} {\mathbf{G}}_{1} {\mathbf{Φ}}_{1}+{F_{\text{S}}} {\mathbf{T}}_{2}-{\mathbf{R}}_{2} \end{displaymath} \begin{displaymath} -{\mathbf{G}}_{3} {\mathbf{Φ}}_{3}={\mathbf{Ψ}}_{2} {\mathbf{G}}_{2} {\mathbf{Φ}}_{2}+{F_{S}} {\mathbf{T}}_{3}-{\mathbf{R}}_{3} +{\mathbf{G}}_{3} {\mathbf{Φ}}_{3}={\mathbf{Ψ}}_{2} {\mathbf{G}}_{2} {\mathbf{Φ}}_{2}+{F_{\text{S}}} {\mathbf{T}}_{3}-{\mathbf{R}}_{3} \end{displaymath} ... \begin{displaymath} -{\mathbf{G}}_{n-2} {\mathbf{Φ}}_{n-2}={\mathbf{Ψ}}_{n-3} {\mathbf{G}}_{n-3} {\mathbf{Φ}}_{n-3}+{F_{S}} {\mathbf{T}}_{n-2}-{\mathbf{R}}_{n-2} +{\mathbf{G}}_{n-2} {\mathbf{Φ}}_{n-2}={\mathbf{Ψ}}_{n-3} {\mathbf{G}}_{n-3} {\mathbf{Φ}}_{n-3}+{F_{\text{S}}} {\mathbf{T}}_{n-2}-{\mathbf{R}}_{n-2} \end{displaymath} \begin{displaymath} -{\mathbf{G}}_{n-1} {\mathbf{Φ}}_{n-1}={\mathbf{Ψ}}_{n-2} {\mathbf{G}}_{n-2} {\mathbf{Φ}}_{n-2}+{F_{S}} {\mathbf{T}}_{n-1}-{\mathbf{R}}_{n-1} +{\mathbf{G}}_{n-1} {\mathbf{Φ}}_{n-1}={\mathbf{Ψ}}_{n-2} {\mathbf{G}}_{n-2} {\mathbf{Φ}}_{n-2}+{F_{\text{S}}} {\mathbf{T}}_{n-1}-{\mathbf{R}}_{n-1} \end{displaymath} \begin{displaymath} -{\mathbf{G}}_{n} {\mathbf{Φ}}_{n}={\mathbf{Ψ}}_{n-1} {\mathbf{G}}_{n-1} {\mathbf{Φ}}_{n-1}+{F_{S}} {\mathbf{T}}_{n}-{\mathbf{R}}_{n} +{\mathbf{G}}_{n} {\mathbf{Φ}}_{n}={\mathbf{Ψ}}_{n-1} {\mathbf{G}}_{n-1} {\mathbf{Φ}}_{n-1}+{F_{\text{S}}} {\mathbf{T}}_{n}-{\mathbf{R}}_{n} \end{displaymath} Applying \hyperref[assumpES]{A: Edge-Slices}, which says that ${\mathbf{G}}_{0}$ and ${\mathbf{G}}_{n}$ are zero, results in the following special cases: Equation (8) for the first slice: \begin{displaymath} -{\mathbf{G}}_{1} {\mathbf{Φ}}_{1}={F_{S}} {\mathbf{T}}_{1}-{\mathbf{R}}_{1} +{\mathbf{G}}_{1} {\mathbf{Φ}}_{1}={F_{\text{S}}} {\mathbf{T}}_{1}-{\mathbf{R}}_{1} \end{displaymath} and Equation (9) for the $n$th slice: \begin{displaymath} --\left(\frac{{F_{S}} {\mathbf{T}}_{n}-{\mathbf{R}}_{n}}{{\mathbf{Ψ}}_{n-1}}\right)={\mathbf{G}}_{n-1} {\mathbf{Φ}}_{n-1} +-\left(\frac{{F_{\text{S}}} {\mathbf{T}}_{n}-{\mathbf{R}}_{n}}{{\mathbf{Ψ}}_{n-1}}\right)={\mathbf{G}}_{n-1} {\mathbf{Φ}}_{n-1} \end{displaymath} Substituting Equation (8) into Equation (4) yields Equation (10): \begin{displaymath} -{\mathbf{G}}_{2} {\mathbf{Φ}}_{2}={\mathbf{Ψ}}_{1} \left({F_{S}} {\mathbf{T}}_{1}-{\mathbf{R}}_{1}\right)+{F_{S}} {\mathbf{T}}_{2}-{\mathbf{R}}_{2} +{\mathbf{G}}_{2} {\mathbf{Φ}}_{2}={\mathbf{Ψ}}_{1} \left({F_{\text{S}}} {\mathbf{T}}_{1}-{\mathbf{R}}_{1}\right)+{F_{\text{S}}} {\mathbf{T}}_{2}-{\mathbf{R}}_{2} \end{displaymath} which can be substituted into Equation (5) to get Equation (11): \begin{displaymath} -{\mathbf{G}}_{3} {\mathbf{Φ}}_{3}={\mathbf{Ψ}}_{2} \left({\mathbf{Ψ}}_{1} \left({F_{S}} {\mathbf{T}}_{1}-{\mathbf{R}}_{1}\right)+{F_{S}} {\mathbf{T}}_{2}-{\mathbf{R}}_{2}\right)+{F_{S}} {\mathbf{T}}_{3}-{\mathbf{R}}_{3} +{\mathbf{G}}_{3} {\mathbf{Φ}}_{3}={\mathbf{Ψ}}_{2} \left({\mathbf{Ψ}}_{1} \left({F_{\text{S}}} {\mathbf{T}}_{1}-{\mathbf{R}}_{1}\right)+{F_{\text{S}}} {\mathbf{T}}_{2}-{\mathbf{R}}_{2}\right)+{F_{\text{S}}} {\mathbf{T}}_{3}-{\mathbf{R}}_{3} \end{displaymath} and so on until Equation (12) is obtained from Equation (7): \begin{displaymath} -{\mathbf{G}}_{n-1} {\mathbf{Φ}}_{n-1}={\mathbf{Ψ}}_{n-2} \left({\mathbf{Ψ}}_{n-3} \left({\mathbf{Ψ}}_{1} \left({F_{S}} {\mathbf{T}}_{1}-{\mathbf{R}}_{1}\right)+{F_{S}} {\mathbf{T}}_{2}-{\mathbf{R}}_{2}\right)+{F_{S}} {\mathbf{T}}_{n-2}-{\mathbf{R}}_{n-2}\right)+{F_{S}} {\mathbf{T}}_{n-1}-{\mathbf{R}}_{n-1} +{\mathbf{G}}_{n-1} {\mathbf{Φ}}_{n-1}={\mathbf{Ψ}}_{n-2} \left({\mathbf{Ψ}}_{n-3} \left({\mathbf{Ψ}}_{1} \left({F_{\text{S}}} {\mathbf{T}}_{1}-{\mathbf{R}}_{1}\right)+{F_{\text{S}}} {\mathbf{T}}_{2}-{\mathbf{R}}_{2}\right)+{F_{\text{S}}} {\mathbf{T}}_{n-2}-{\mathbf{R}}_{n-2}\right)+{F_{\text{S}}} {\mathbf{T}}_{n-1}-{\mathbf{R}}_{n-1} \end{displaymath} Equation (9) can then be substituted into the left-hand side of Equation (12), resulting in: \begin{displaymath} --\left(\frac{{F_{S}} {\mathbf{T}}_{n}-{\mathbf{R}}_{n}}{{\mathbf{Ψ}}_{n-1}}\right)={\mathbf{Ψ}}_{n-2} \left({\mathbf{Ψ}}_{n-3} \left({\mathbf{Ψ}}_{1} \left({F_{S}} {\mathbf{T}}_{1}-{\mathbf{R}}_{1}\right)+{F_{S}} {\mathbf{T}}_{2}-{\mathbf{R}}_{2}\right)+{F_{S}} {\mathbf{T}}_{n-2}-{\mathbf{R}}_{n-2}\right)+{F_{S}} {\mathbf{T}}_{n-1}-{\mathbf{R}}_{n-1} +-\left(\frac{{F_{\text{S}}} {\mathbf{T}}_{n}-{\mathbf{R}}_{n}}{{\mathbf{Ψ}}_{n-1}}\right)={\mathbf{Ψ}}_{n-2} \left({\mathbf{Ψ}}_{n-3} \left({\mathbf{Ψ}}_{1} \left({F_{\text{S}}} {\mathbf{T}}_{1}-{\mathbf{R}}_{1}\right)+{F_{\text{S}}} {\mathbf{T}}_{2}-{\mathbf{R}}_{2}\right)+{F_{\text{S}}} {\mathbf{T}}_{n-2}-{\mathbf{R}}_{n-2}\right)+{F_{\text{S}}} {\mathbf{T}}_{n-1}-{\mathbf{R}}_{n-1} \end{displaymath} This can be rearranged by multiplying boths sides by ${\mathbf{Ψ}}_{n-1}$ and then distributing the multiplication of each $\mathbf{Ψ}$ over addition to obtain: \begin{displaymath} --\left({F_{S}} {\mathbf{T}}_{n}-{\mathbf{R}}_{n}\right)={\mathbf{Ψ}}_{n-1} {\mathbf{Ψ}}_{n-2} {\mathbf{Ψ}}_{1} \left({F_{S}} {\mathbf{T}}_{1}-{\mathbf{R}}_{1}\right)+{\mathbf{Ψ}}_{n-1} {\mathbf{Ψ}}_{n-2} {\mathbf{Ψ}}_{2} \left({F_{S}} {\mathbf{T}}_{2}-{\mathbf{R}}_{2}\right)+{\mathbf{Ψ}}_{n-1} \left({F_{S}} {\mathbf{T}}_{n-1}-{\mathbf{R}}_{n-1}\right) +-\left({F_{\text{S}}} {\mathbf{T}}_{n}-{\mathbf{R}}_{n}\right)={\mathbf{Ψ}}_{n-1} {\mathbf{Ψ}}_{n-2} {\mathbf{Ψ}}_{1} \left({F_{\text{S}}} {\mathbf{T}}_{1}-{\mathbf{R}}_{1}\right)+{\mathbf{Ψ}}_{n-1} {\mathbf{Ψ}}_{n-2} {\mathbf{Ψ}}_{2} \left({F_{\text{S}}} {\mathbf{T}}_{2}-{\mathbf{R}}_{2}\right)+{\mathbf{Ψ}}_{n-1} \left({F_{\text{S}}} {\mathbf{T}}_{n-1}-{\mathbf{R}}_{n-1}\right) \end{displaymath} The multiplication of the $\mathbf{Ψ}$ terms can be further distributed over the subtractions, resulting in the equation having terms that each either contain an $\mathbf{R}$ or a $\mathbf{T}$. The equation can then be rearranged so terms containing an $\mathbf{R}$ are on one side of the equality, and terms containing a $\mathbf{T}$ are on the other. The multiplication by the factor of safety is common to all of the $\mathbf{T}$ terms, and thus can be factored out, resulting in: \begin{displaymath} -{F_{S}} \left({\mathbf{Ψ}}_{n-1} {\mathbf{Ψ}}_{n-2} {\mathbf{Ψ}}_{1} {\mathbf{T}}_{1}+{\mathbf{Ψ}}_{n-1} {\mathbf{Ψ}}_{n-2} {\mathbf{Ψ}}_{2} {\mathbf{T}}_{2}+{\mathbf{Ψ}}_{n-1} {\mathbf{T}}_{n-1}+{\mathbf{T}}_{n}\right)={\mathbf{Ψ}}_{n-1} {\mathbf{Ψ}}_{n-2} {\mathbf{Ψ}}_{1} {\mathbf{R}}_{1}+{\mathbf{Ψ}}_{n-1} {\mathbf{Ψ}}_{n-2} {\mathbf{Ψ}}_{2} {\mathbf{R}}_{2}+{\mathbf{Ψ}}_{n-1} {\mathbf{R}}_{n-1}+{\mathbf{R}}_{n} +{F_{\text{S}}} \left({\mathbf{Ψ}}_{n-1} {\mathbf{Ψ}}_{n-2} {\mathbf{Ψ}}_{1} {\mathbf{T}}_{1}+{\mathbf{Ψ}}_{n-1} {\mathbf{Ψ}}_{n-2} {\mathbf{Ψ}}_{2} {\mathbf{T}}_{2}+{\mathbf{Ψ}}_{n-1} {\mathbf{T}}_{n-1}+{\mathbf{T}}_{n}\right)={\mathbf{Ψ}}_{n-1} {\mathbf{Ψ}}_{n-2} {\mathbf{Ψ}}_{1} {\mathbf{R}}_{1}+{\mathbf{Ψ}}_{n-1} {\mathbf{Ψ}}_{n-2} {\mathbf{Ψ}}_{2} {\mathbf{R}}_{2}+{\mathbf{Ψ}}_{n-1} {\mathbf{R}}_{n-1}+{\mathbf{R}}_{n} \end{displaymath} Isolating the factor of safety on the left-hand side and using compact notation for the products and sums yields Equation (13), which can also be seen in \hyperref[IM:fctSfty]{IM: fctSfty}: \begin{displaymath} -{F_{S}}=\frac{\displaystyle\sum_{i=1}^{n-1}{{\mathbf{R}}_{i} \displaystyle\prod_{v=i}^{n-1}{{\mathbf{Ψ}}_{v}}}+{\mathbf{R}}_{n}}{\displaystyle\sum_{i=1}^{n-1}{{\mathbf{T}}_{i} \displaystyle\prod_{v=i}^{n-1}{{\mathbf{Ψ}}_{v}}}+{\mathbf{T}}_{n}} +{F_{\text{S}}}=\frac{\displaystyle\sum_{i=1}^{n-1}{{\mathbf{R}}_{i} \displaystyle\prod_{v=i}^{n-1}{{\mathbf{Ψ}}_{v}}}+{\mathbf{R}}_{n}}{\displaystyle\sum_{i=1}^{n-1}{{\mathbf{T}}_{i} \displaystyle\prod_{v=i}^{n-1}{{\mathbf{Ψ}}_{v}}}+{\mathbf{T}}_{n}} \end{displaymath} -${F_{S}}$ depends on the unknowns $λ$ (\hyperref[IM:nrmShrFor]{IM: nrmShrFor}) and $\mathbf{G}$ (\hyperref[IM:intsliceFs]{IM: intsliceFs}). +${F_{\text{S}}}$ depends on the unknowns $λ$ (\hyperref[IM:nrmShrFor]{IM: nrmShrFor}) and $\mathbf{G}$ (\hyperref[IM:intsliceFs]{IM: intsliceFs}). \par~ \noindent \begin{minipage}{\textwidth} @@ -1897,7 +1897,7 @@ \subsubsection{Instance Models} \\ \midrule \\ Label & Normal and shear force proportionality constant \\ \midrule \\ -Input & ${\mathbf{x}_{slope}}$, ${\mathbf{y}_{slope}}$, ${\mathbf{y}_{wt}}$, ${γ_{w}}$, ${\mathbf{x}_{slip}}$, ${\mathbf{y}_{slip}}$, $const_f$ +Input & ${\mathbf{x}_{\text{slope}}}$, ${\mathbf{y}_{\text{slope}}}$, ${\mathbf{y}_{\text{wt}}}$, ${γ_{w}}$, ${\mathbf{x}_{\text{slip}}}$, ${\mathbf{y}_{\text{slip}}}$, $const_f$ \\ \midrule \\ Output & $λ$ \\ \midrule \\ @@ -1906,17 +1906,17 @@ \subsubsection{Instance Models} Output Constraints & \\ \midrule \\ Equation & \begin{displaymath} - λ=\frac{\displaystyle\sum_{i=1}^{n}{{\mathbf{C}_{num,i}}}}{\displaystyle\sum_{i=1}^{n}{{\mathbf{C}_{den,i}}}} + λ=\frac{\displaystyle\sum_{i=1}^{n}{{\mathbf{C}_{\text{num},i}}}}{\displaystyle\sum_{i=1}^{n}{{\mathbf{C}_{\text{den},i}}}} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} \item{$λ$ is the proportionality constant (Unitless)} - \item{${\mathbf{C}_{num}}$ is the proportionality constant numerator (N)} + \item{${\mathbf{C}_{\text{num}}}$ is the proportionality constant numerator (N)} \item{$i$ is the index (Unitless)} - \item{${\mathbf{C}_{den}}$ is the proportionality constant denominator (N)} + \item{${\mathbf{C}_{\text{den}}}$ is the proportionality constant denominator (N)} \end{symbDescription} \\ \midrule \\ -Notes & ${\mathbf{C}_{num}}$ is defined in \hyperref[IM:nrmShrForNum]{IM: nrmShrForNum} and ${\mathbf{C}_{den}}$ is defined in \hyperref[IM:nrmShrForDen]{IM: nrmShrForDen}. +Notes & ${\mathbf{C}_{\text{num}}}$ is defined in \hyperref[IM:nrmShrForNum]{IM: nrmShrForNum} and ${\mathbf{C}_{\text{den}}}$ is defined in \hyperref[IM:nrmShrForDen]{IM: nrmShrForDen}. \\ \midrule \\ Source & \cite{chen2005} and \cite{karchewski2012} \\ \midrule \\ @@ -1928,21 +1928,21 @@ \subsubsection{Instance Models} \label{IM:nrmShrForDeriv} From the moment equilibrium of \hyperref[GD:momentEql]{GD: momentEql} with the primary assumption for the Morgenstern-Price method of \hyperref[assumpINSFL]{A: Interslice-Norm-Shear-Forces-Linear} and associated definition \hyperref[GD:normShrR]{GD: normShrR}, Equation (14) can be derived: \begin{displaymath} -0=-{\mathbf{G}}_{i} \left({\mathbf{h}_{z,i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{G}}_{i-1} \left({\mathbf{h}_{z,i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)-{\mathbf{H}}_{i} \left(\frac{1}{3} {\mathbf{h}_{z,w,i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{H}}_{i-1} \left(\frac{1}{3} {\mathbf{h}_{z,w,i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+λ \frac{{\mathbf{b}}_{i}}{2} \left({\mathbf{G}}_{i} {\mathbf{f}}_{i}+{\mathbf{G}}_{i-1} {\mathbf{f}}_{i-1}\right)+\frac{-{K_{c}} {\mathbf{W}}_{i} {\mathbf{h}}_{i}}{2}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right) {\mathbf{h}}_{i}+{\mathbf{Q}}_{i} \sin\left({\mathbf{ω}}_{i}\right) {\mathbf{h}}_{i} +0=-{\mathbf{G}}_{i} \left({\mathbf{h}_{z,i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{G}}_{i-1} \left({\mathbf{h}_{z,i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)-{\mathbf{H}}_{i} \left(\frac{1}{3} {\mathbf{h}_{\text{z,w},i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{H}}_{i-1} \left(\frac{1}{3} {\mathbf{h}_{\text{z,w},i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+λ \frac{{\mathbf{b}}_{i}}{2} \left({\mathbf{G}}_{i} {\mathbf{f}}_{i}+{\mathbf{G}}_{i-1} {\mathbf{f}}_{i-1}\right)+\frac{-{K_{c}} {\mathbf{W}}_{i} {\mathbf{h}}_{i}}{2}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right) {\mathbf{h}}_{i}+{\mathbf{Q}}_{i} \sin\left({\mathbf{ω}}_{i}\right) {\mathbf{h}}_{i} \end{displaymath} Rearranging the equation in terms of $λ$ leads to Equation (15): \begin{displaymath} -λ=\frac{-{\mathbf{G}}_{i} \left({\mathbf{h}_{z,i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{G}}_{i-1} \left({\mathbf{h}_{z,i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)-{\mathbf{H}}_{i} \left(\frac{1}{3} {\mathbf{h}_{z,w,i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{H}}_{i-1} \left(\frac{1}{3} {\mathbf{h}_{z,w,i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+\frac{-{K_{c}} {\mathbf{W}}_{i} {\mathbf{h}}_{i}}{2}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right) {\mathbf{h}}_{i}+{\mathbf{Q}}_{i} \sin\left({\mathbf{ω}}_{i}\right) {\mathbf{h}}_{i}}{-\frac{{\mathbf{b}}_{i}}{2} \left({\mathbf{G}}_{i} {\mathbf{f}}_{i}+{\mathbf{G}}_{i-1} {\mathbf{f}}_{i-1}\right)} +λ=\frac{-{\mathbf{G}}_{i} \left({\mathbf{h}_{z,i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{G}}_{i-1} \left({\mathbf{h}_{z,i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)-{\mathbf{H}}_{i} \left(\frac{1}{3} {\mathbf{h}_{\text{z,w},i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{H}}_{i-1} \left(\frac{1}{3} {\mathbf{h}_{\text{z,w},i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+\frac{-{K_{c}} {\mathbf{W}}_{i} {\mathbf{h}}_{i}}{2}+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right) {\mathbf{h}}_{i}+{\mathbf{Q}}_{i} \sin\left({\mathbf{ω}}_{i}\right) {\mathbf{h}}_{i}}{-\frac{{\mathbf{b}}_{i}}{2} \left({\mathbf{G}}_{i} {\mathbf{f}}_{i}+{\mathbf{G}}_{i-1} {\mathbf{f}}_{i-1}\right)} \end{displaymath} This equation can be simplified by applying assumptions \hyperref[assumpSF]{A: Seismic-Force} and \hyperref[assumpSL]{A: Surface-Load}, which state that the seismic and external forces, respectively, are zero: \begin{displaymath} -λ=\frac{-{\mathbf{G}}_{i} \left({\mathbf{h}_{z,i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{G}}_{i-1} \left({\mathbf{h}_{z,i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)-{\mathbf{H}}_{i} \left(\frac{1}{3} {\mathbf{h}_{z,w,i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{H}}_{i-1} \left(\frac{1}{3} {\mathbf{h}_{z,w,i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right) {\mathbf{h}}_{i}}{-\frac{{\mathbf{b}}_{i}}{2} \left({\mathbf{G}}_{i} {\mathbf{f}}_{i}+{\mathbf{G}}_{i-1} {\mathbf{f}}_{i-1}\right)} +λ=\frac{-{\mathbf{G}}_{i} \left({\mathbf{h}_{z,i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{G}}_{i-1} \left({\mathbf{h}_{z,i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)-{\mathbf{H}}_{i} \left(\frac{1}{3} {\mathbf{h}_{\text{z,w},i}}+\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{H}}_{i-1} \left(\frac{1}{3} {\mathbf{h}_{\text{z,w},i-1}}-\frac{{\mathbf{b}}_{i}}{2} \tan\left({\mathbf{α}}_{i}\right)\right)+{\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right) {\mathbf{h}}_{i}}{-\frac{{\mathbf{b}}_{i}}{2} \left({\mathbf{G}}_{i} {\mathbf{f}}_{i}+{\mathbf{G}}_{i-1} {\mathbf{f}}_{i-1}\right)} \end{displaymath} Taking the summation of all slices, and applying \hyperref[assumpES]{A: Edge-Slices} to set ${\mathbf{G}}_{0}$, ${\mathbf{G}}_{n}$, ${\mathbf{H}}_{0}$, and ${\mathbf{H}}_{n}$ equal to zero, a general equation for the proportionality constant $λ$ is developed in Equation (16), which combines \hyperref[IM:nrmShrFor]{IM: nrmShrFor}, \hyperref[IM:nrmShrForNum]{IM: nrmShrForNum}, and \hyperref[IM:nrmShrForDen]{IM: nrmShrForDen}: \begin{displaymath} λ=\frac{\displaystyle\sum_{i=1}^{n}{{\mathbf{b}}_{i} \left({{\mathbf{F}_{x}}^{G}}+{{\mathbf{F}_{x}}^{H}}\right) \tan\left({\mathbf{α}}_{i}\right)+{\mathbf{h}}_{i} -2 {\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right)}}{\displaystyle\sum_{i=1}^{n}{{\mathbf{b}}_{i} \left({\mathbf{G}}_{i} {\mathbf{f}}_{i}+{\mathbf{G}}_{i-1} {\mathbf{f}}_{i-1}\right)}} \end{displaymath} -Equation (16) for $λ$ is a function of the unknown interslice normal forces $\mathbf{G}$ (\hyperref[IM:intsliceFs]{IM: intsliceFs}) which itself depends on the unknown factor of safety ${F_{S}}$ (\hyperref[IM:fctSfty]{IM: fctSfty}). +Equation (16) for $λ$ is a function of the unknown interslice normal forces $\mathbf{G}$ (\hyperref[IM:intsliceFs]{IM: intsliceFs}) which itself depends on the unknown factor of safety ${F_{\text{S}}}$ (\hyperref[IM:fctSfty]{IM: fctSfty}). \par~ \noindent \begin{minipage}{\textwidth} @@ -1953,24 +1953,24 @@ \subsubsection{Instance Models} \\ \midrule \\ Label & Normal and shear force proportionality constant numerator \\ \midrule \\ -Input & ${\mathbf{x}_{slope}}$, ${\mathbf{y}_{slope}}$, ${\mathbf{y}_{wt}}$, ${γ_{w}}$, ${\mathbf{x}_{slip}}$, ${\mathbf{y}_{slip}}$ +Input & ${\mathbf{x}_{\text{slope}}}$, ${\mathbf{y}_{\text{slope}}}$, ${\mathbf{y}_{\text{wt}}}$, ${γ_{w}}$, ${\mathbf{x}_{\text{slip}}}$, ${\mathbf{y}_{\text{slip}}}$ \\ \midrule \\ -Output & ${\mathbf{C}_{num}}$ +Output & ${\mathbf{C}_{\text{num}}}$ \\ \midrule \\ Input Constraints & \\ \midrule \\ Output Constraints & \\ \midrule \\ Equation & \begin{displaymath} - {\mathbf{C}_{num,i}}=\begin{cases} - {\mathbf{b}}_{1} \left({\mathbf{G}}_{1}+{\mathbf{H}}_{1}\right) \tan\left({\mathbf{α}}_{1}\right), & i=1\\ + {\mathbf{C}_{\text{num},i}}=\begin{cases} + {\mathbf{b}}_{1} \left({\mathbf{G}}_{1}+{\mathbf{H}}_{1}\right) \tan\left({\mathbf{α}}_{1}\right), & i=1\\ {\mathbf{b}}_{i} \left({{\mathbf{F}_{x}}^{G}}+{{\mathbf{F}_{x}}^{H}}\right) \tan\left({\mathbf{α}}_{i}\right)+\mathbf{h} -2 {\mathbf{U}_{t,i}} \sin\left({\mathbf{β}}_{i}\right), & 2\leq{}i\leq{}n-1\\ {\mathbf{b}}_{n} \left({\mathbf{G}}_{n-1}+{\mathbf{H}}_{n-1}\right) \tan\left({\mathbf{α}}_{n-1}\right), & i=n - \end{cases} + \end{cases} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} - \item{${\mathbf{C}_{num}}$ is the proportionality constant numerator (N)} + \item{${\mathbf{C}_{\text{num}}}$ is the proportionality constant numerator (N)} \item{$i$ is the index (Unitless)} \item{$\mathbf{b}$ is the base width of slices (m)} \item{$\mathbf{G}$ is the interslice normal forces ($\frac{\text{N}}{\text{m}}$)} @@ -1994,7 +1994,7 @@ \subsubsection{Instance Models} \end{minipage} \paragraph{} \label{IM:nrmShrForNumDeriv} -See \hyperref[IM:nrmShrFor]{IM: nrmShrFor} for the derivation of ${\mathbf{C}_{num}}$. +See \hyperref[IM:nrmShrFor]{IM: nrmShrFor} for the derivation of ${\mathbf{C}_{\text{num}}}$. \par~ \noindent \begin{minipage}{\textwidth} @@ -2005,24 +2005,24 @@ \subsubsection{Instance Models} \\ \midrule \\ Label & Normal and shear force proportionality constant denominator \\ \midrule \\ -Input & ${\mathbf{x}_{slip}}$, $const_f$ +Input & ${\mathbf{x}_{\text{slip}}}$, $const_f$ \\ \midrule \\ -Output & ${\mathbf{C}_{den}}$ +Output & ${\mathbf{C}_{\text{den}}}$ \\ \midrule \\ Input Constraints & \\ \midrule \\ Output Constraints & \\ \midrule \\ Equation & \begin{displaymath} - {\mathbf{C}_{den,i}}=\begin{cases} - {\mathbf{b}}_{1} {\mathbf{f}}_{1} {\mathbf{G}}_{1}, & i=1\\ + {\mathbf{C}_{\text{den},i}}=\begin{cases} + {\mathbf{b}}_{1} {\mathbf{f}}_{1} {\mathbf{G}}_{1}, & i=1\\ {\mathbf{b}}_{i} \left({\mathbf{f}}_{i} {\mathbf{G}}_{i}+{\mathbf{f}}_{i-1} {\mathbf{G}}_{i-1}\right), & 2\leq{}i\leq{}n-1\\ {\mathbf{b}}_{n} {\mathbf{G}}_{n-1} {\mathbf{f}}_{n-1}, & i=1 - \end{cases} + \end{cases} \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} - \item{${\mathbf{C}_{den}}$ is the proportionality constant denominator (N)} + \item{${\mathbf{C}_{\text{den}}}$ is the proportionality constant denominator (N)} \item{$i$ is the index (Unitless)} \item{$\mathbf{b}$ is the base width of slices (m)} \item{$\mathbf{f}$ is the interslice normal to shear force ratio variation function (Unitless)} @@ -2040,7 +2040,7 @@ \subsubsection{Instance Models} \end{minipage} \paragraph{} \label{IM:nrmShrForDenDeriv} -See \hyperref[IM:nrmShrFor]{IM: nrmShrFor} for the derivation of ${\mathbf{C}_{den}}$. +See \hyperref[IM:nrmShrFor]{IM: nrmShrFor} for the derivation of ${\mathbf{C}_{\text{den}}}$. \par~ \noindent \begin{minipage}{\textwidth} @@ -2051,7 +2051,7 @@ \subsubsection{Instance Models} \\ \midrule \\ Label & Interslice normal forces \\ \midrule \\ -Input & ${\mathbf{x}_{slope}}$, ${\mathbf{y}_{slope}}$, ${\mathbf{y}_{wt}}$, $c'$, $φ'$, ${γ_{dry}}$, ${γ_{Sat}}$, ${γ_{w}}$, ${\mathbf{x}_{slip}}$, ${\mathbf{y}_{slip}}$, $const_f$ +Input & ${\mathbf{x}_{\text{slope}}}$, ${\mathbf{y}_{\text{slope}}}$, ${\mathbf{y}_{\text{wt}}}$, $c'$, $φ'$, ${γ_{\text{dry}}}$, ${γ_{\text{Sat}}}$, ${γ_{w}}$, ${\mathbf{x}_{\text{slip}}}$, ${\mathbf{y}_{\text{slip}}}$, $const_f$ \\ \midrule \\ Output & $\mathbf{G}$ \\ \midrule \\ @@ -2061,8 +2061,8 @@ \subsubsection{Instance Models} \\ \midrule \\ Equation & \begin{displaymath} {\mathbf{G}}_{i}=\begin{cases} - \frac{{F_{S}} {\mathbf{T}}_{1}-{\mathbf{R}}_{1}}{{\mathbf{Φ}}_{1}}, & i=1\\ -\frac{{\mathbf{Ψ}}_{i-1} {\mathbf{G}}_{i-1}+{F_{S}} {\mathbf{T}}_{i}-{\mathbf{R}}_{i}}{{\mathbf{Φ}}_{i}}, & 2\leq{}i\leq{}n-1\\ + \frac{{F_{\text{S}}} {\mathbf{T}}_{1}-{\mathbf{R}}_{1}}{{\mathbf{Φ}}_{1}}, & i=1\\ +\frac{{\mathbf{Ψ}}_{i-1} {\mathbf{G}}_{i-1}+{F_{\text{S}}} {\mathbf{T}}_{i}-{\mathbf{R}}_{i}}{{\mathbf{Φ}}_{i}}, & 2\leq{}i\leq{}n-1\\ 0, & i=0\lor{}i=n \end{cases} \end{displaymath} @@ -2070,7 +2070,7 @@ \subsubsection{Instance Models} Description & \begin{symbDescription} \item{$\mathbf{G}$ is the interslice normal forces ($\frac{\text{N}}{\text{m}}$)} \item{$i$ is the index (Unitless)} - \item{${F_{S}}$ is the factor of safety (Unitless)} + \item{${F_{\text{S}}}$ is the factor of safety (Unitless)} \item{$\mathbf{T}$ is the mobilized shear forces without the influence of interslice forces ($\frac{\text{N}}{\text{m}}$)} \item{$\mathbf{R}$ is the resistive shear forces without the influence of interslice forces ($\frac{\text{N}}{\text{m}}$)} \item{$\mathbf{Φ}$ is the first function for incorporating interslice forces into shear force (Unitless)} @@ -2090,13 +2090,13 @@ \subsubsection{Instance Models} \label{IM:intsliceFsDeriv} This derivation is identical to the derivation for \hyperref[IM:fctSfty]{IM: fctSfty} up until Equation (3) shown again below: \begin{displaymath} -{\mathbf{G}}_{i} {\mathbf{Φ}}_{i}={\mathbf{Ψ}}_{i-1} {\mathbf{G}}_{i-1} {\mathbf{Φ}}_{i-1}+{F_{S}} {\mathbf{T}}_{i}-{\mathbf{R}}_{i} +{\mathbf{G}}_{i} {\mathbf{Φ}}_{i}={\mathbf{Ψ}}_{i-1} {\mathbf{G}}_{i-1} {\mathbf{Φ}}_{i-1}+{F_{\text{S}}} {\mathbf{T}}_{i}-{\mathbf{R}}_{i} \end{displaymath} A simple rearrangement of Equation (3) leads to Equation (17), also seen in \hyperref[IM:intsliceFs]{IM: intsliceFs}: \begin{displaymath} -{\mathbf{G}}_{i}=\frac{{\mathbf{Ψ}}_{i-1} {\mathbf{G}}_{i-1}+{F_{S}} {\mathbf{T}}_{i}-{\mathbf{R}}_{i}}{{\mathbf{Φ}}_{i}} +{\mathbf{G}}_{i}=\frac{{\mathbf{Ψ}}_{i-1} {\mathbf{G}}_{i-1}+{F_{\text{S}}} {\mathbf{T}}_{i}-{\mathbf{R}}_{i}}{{\mathbf{Φ}}_{i}} \end{displaymath} -The cases shown in \hyperref[IM:intsliceFs]{IM: intsliceFs} for when $i=0$, $i=1$, or $i=n$ are derived by applying \hyperref[assumpES]{A: Edge-Slices}, which says that ${\mathbf{G}}_{0}$ and ${\mathbf{G}}_{n}$ are zero, to Equation (17). $\mathbf{G}$ depends on the unknowns ${F_{S}}$ (\hyperref[IM:fctSfty]{IM: fctSfty}) and $λ$ (\hyperref[IM:nrmShrFor]{IM: nrmShrFor}). +The cases shown in \hyperref[IM:intsliceFs]{IM: intsliceFs} for when $i=0$, $i=1$, or $i=n$ are derived by applying \hyperref[assumpES]{A: Edge-Slices}, which says that ${\mathbf{G}}_{0}$ and ${\mathbf{G}}_{n}$ are zero, to Equation (17). $\mathbf{G}$ depends on the unknowns ${F_{\text{S}}}$ (\hyperref[IM:fctSfty]{IM: fctSfty}) and $λ$ (\hyperref[IM:nrmShrFor]{IM: nrmShrFor}). \par~ \noindent \begin{minipage}{\textwidth} @@ -2107,34 +2107,34 @@ \subsubsection{Instance Models} \\ \midrule \\ Label & Critical slip surface identification \\ \midrule \\ -Input & ${\mathbf{x}_{slope}}$, ${\mathbf{y}_{slope}}$, ${\mathbf{x}_{wt}}$, ${\mathbf{y}_{wt}}$, $c'$, $φ'$, ${γ_{dry}}$, ${γ_{Sat}}$, ${γ_{w}}$, $const_f$ +Input & ${\mathbf{x}_{\text{slope}}}$, ${\mathbf{y}_{\text{slope}}}$, ${\mathbf{x}_{\text{wt}}}$, ${\mathbf{y}_{\text{wt}}}$, $c'$, $φ'$, ${γ_{\text{dry}}}$, ${γ_{\text{Sat}}}$, ${γ_{w}}$, $const_f$ \\ \midrule \\ -Output & ${{F_{S}}^{min}}$ +Output & ${{F_{\text{S}}}^{\text{min}}}$ \\ \midrule \\ Input Constraints & \\ \midrule \\ Output Constraints & \\ \midrule \\ Equation & \begin{displaymath} - {{F_{S}}^{min}}=Υ\left({\mathbf{x}_{slope}},{\mathbf{y}_{slope}},{\mathbf{x}_{wt}},{\mathbf{y}_{wt}},c',φ',{γ_{dry}},{γ_{Sat}},{γ_{w}},const_f\right) + {{F_{\text{S}}}^{\text{min}}}=Υ\left({\mathbf{x}_{\text{slope}}},{\mathbf{y}_{\text{slope}}},{\mathbf{x}_{\text{wt}}},{\mathbf{y}_{\text{wt}}},c',φ',{γ_{\text{dry}}},{γ_{\text{Sat}}},{γ_{w}},const_f\right) \end{displaymath} \\ \midrule \\ Description & \begin{symbDescription} - \item{${{F_{S}}^{min}}$ is the minimum factor of safety (Unitless)} + \item{${{F_{\text{S}}}^{\text{min}}}$ is the minimum factor of safety (Unitless)} \item{$Υ$ is the minimization function (Unitless)} - \item{${\mathbf{x}_{slope}}$ is the x-coordinates of the slope (m)} - \item{${\mathbf{y}_{slope}}$ is the y-coordinates of the slope (m)} - \item{${\mathbf{x}_{wt}}$ is the x-coordinates of the water table (m)} - \item{${\mathbf{y}_{wt}}$ is the y-coordinates of the water table (m)} + \item{${\mathbf{x}_{\text{slope}}}$ is the x-coordinates of the slope (m)} + \item{${\mathbf{y}_{\text{slope}}}$ is the y-coordinates of the slope (m)} + \item{${\mathbf{x}_{\text{wt}}}$ is the x-coordinates of the water table (m)} + \item{${\mathbf{y}_{\text{wt}}}$ is the y-coordinates of the water table (m)} \item{$c'$ is the effective cohesion (Pa)} \item{$φ'$ is the effective angle of friction (${}^{\circ}$)} - \item{${γ_{dry}}$ is the soil dry unit weight ($\frac{\text{N}}{\text{m}^{3}}$)} - \item{${γ_{Sat}}$ is the soil saturated unit weight ($\frac{\text{N}}{\text{m}^{3}}$)} + \item{${γ_{\text{dry}}}$ is the soil dry unit weight ($\frac{\text{N}}{\text{m}^{3}}$)} + \item{${γ_{\text{Sat}}}$ is the soil saturated unit weight ($\frac{\text{N}}{\text{m}^{3}}$)} \item{${γ_{w}}$ is the unit weight of water ($\frac{\text{N}}{\text{m}^{3}}$)} \item{$const_f$ is the decision on f (Unitless)} \end{symbDescription} \\ \midrule \\ -Notes & The minimization function must enforce the constraints on the critical slip surface expressed in \hyperref[assumpSSC]{A: Slip-Surface-Concave} and \hyperref[Sec:CorSolProps]{Section: Properties of a Correct Solution}. The sizes of ${\mathbf{x}_{wt}}$ and ${\mathbf{y}_{wt}}$ must be equal and not 1. The sizes of ${\mathbf{x}_{slope}}$ and ${\mathbf{y}_{slope}}$ must be equal and at least 2. The first and last ${\mathbf{x}_{wt}}$ values must be equal to the first and last ${\mathbf{x}_{slope}}$ values. ${\mathbf{x}_{wt}}$ and ${\mathbf{x}_{slope}}$ values must be monotonically increasing. ${{x_{slip}}^{maxExt}}$, ${{x_{slip}}^{maxEtr}}$, ${{x_{slip}}^{minExt}}$, and ${{x_{slip}}^{minEtr}}$ must be between or equal to the minimum and maximum ${\mathbf{x}_{slope}}$ values. ${{y_{slip}}^{max}}$ cannot be below the minimum ${\mathbf{y}_{slope}}$ value. ${{y_{slip}}^{min}}$ cannot be above the maximum ${\mathbf{y}_{slope}}$ value. All $x$ values of ${\mathbf{x}_{cs}},{\mathbf{y}_{cs}}$ must be between ${{x_{slip}}^{minEtr}}$ and ${{x_{slip}}^{maxExt}}$. All $y$ values of ${\mathbf{x}_{cs}},{\mathbf{y}_{cs}}$ must not be below ${{y_{slip}}^{min}}$. For any given vertex in ${\mathbf{x}_{cs}},{\mathbf{y}_{cs}}$ the $y$ value must not exceed the ${\mathbf{y}_{slope}}$ value corresponding to the same $x$ value. The first and last vertices in ${\mathbf{x}_{cs}},{\mathbf{y}_{cs}}$ must each be equal to one of the vertices formed by ${\mathbf{x}_{slope}}$ and ${\mathbf{y}_{slope}}$. The slope between consecutive vertices must be always increasing as $x$ increases. The internal angle between consecutive vertices in ${\mathbf{x}_{cs}},{\mathbf{y}_{cs}}$ must not be below 110 degrees. +Notes & The minimization function must enforce the constraints on the critical slip surface expressed in \hyperref[assumpSSC]{A: Slip-Surface-Concave} and \hyperref[Sec:CorSolProps]{Section: Properties of a Correct Solution}. The sizes of ${\mathbf{x}_{\text{wt}}}$ and ${\mathbf{y}_{\text{wt}}}$ must be equal and not 1. The sizes of ${\mathbf{x}_{\text{slope}}}$ and ${\mathbf{y}_{\text{slope}}}$ must be equal and at least 2. The first and last ${\mathbf{x}_{\text{wt}}}$ values must be equal to the first and last ${\mathbf{x}_{\text{slope}}}$ values. ${\mathbf{x}_{\text{wt}}}$ and ${\mathbf{x}_{\text{slope}}}$ values must be monotonically increasing. ${{x_{\text{slip}}}^{\text{maxExt}}}$, ${{x_{\text{slip}}}^{\text{maxEtr}}}$, ${{x_{\text{slip}}}^{\text{minExt}}}$, and ${{x_{\text{slip}}}^{\text{minEtr}}}$ must be between or equal to the minimum and maximum ${\mathbf{x}_{\text{slope}}}$ values. ${{y_{\text{slip}}}^{\text{max}}}$ cannot be below the minimum ${\mathbf{y}_{\text{slope}}}$ value. ${{y_{\text{slip}}}^{\text{min}}}$ cannot be above the maximum ${\mathbf{y}_{\text{slope}}}$ value. All $x$ values of ${\mathbf{x}_{\text{cs}}}\text{,}{\mathbf{y}_{\text{cs}}}$ must be between ${{x_{\text{slip}}}^{\text{minEtr}}}$ and ${{x_{\text{slip}}}^{\text{maxExt}}}$. All $y$ values of ${\mathbf{x}_{\text{cs}}}\text{,}{\mathbf{y}_{\text{cs}}}$ must not be below ${{y_{\text{slip}}}^{\text{min}}}$. For any given vertex in ${\mathbf{x}_{\text{cs}}}\text{,}{\mathbf{y}_{\text{cs}}}$ the $y$ value must not exceed the ${\mathbf{y}_{\text{slope}}}$ value corresponding to the same $x$ value. The first and last vertices in ${\mathbf{x}_{\text{cs}}}\text{,}{\mathbf{y}_{\text{cs}}}$ must each be equal to one of the vertices formed by ${\mathbf{x}_{\text{slope}}}$ and ${\mathbf{y}_{\text{slope}}}$. The slope between consecutive vertices must be always increasing as $x$ increases. The internal angle between consecutive vertices in ${\mathbf{x}_{\text{cs}}}\text{,}{\mathbf{y}_{\text{cs}}}$ must not be below 110 degrees. \\ \midrule \\ Source & \cite{li2010} \\ \midrule \\ @@ -2153,29 +2153,29 @@ \subsubsection{Data Constraints} \endhead $c'$ & $c'>0$ & $10.0\cdot{}10^{3}$ Pa & 10$\%$ \\ -${{x_{slip}}^{maxEtr}}$ & -- & $20.0$ m & 10$\%$ +${{x_{\text{slip}}}^{\text{maxEtr}}}$ & -- & $20.0$ m & 10$\%$ \\ -${{x_{slip}}^{maxExt}}$ & -- & $100.0$ m & 10$\%$ +${{x_{\text{slip}}}^{\text{maxExt}}}$ & -- & $100.0$ m & 10$\%$ \\ -${{x_{slip}}^{minEtr}}$ & -- & $0.0$ m & 10$\%$ +${{x_{\text{slip}}}^{\text{minEtr}}}$ & -- & $0.0$ m & 10$\%$ \\ -${{x_{slip}}^{minExt}}$ & -- & $50.0$ m & 10$\%$ +${{x_{\text{slip}}}^{\text{minExt}}}$ & -- & $50.0$ m & 10$\%$ \\ -${\mathbf{x}_{slope}}$ & -- & $0.0$ m & 10$\%$ +${\mathbf{x}_{\text{slope}}}$ & -- & $0.0$ m & 10$\%$ \\ -${\mathbf{x}_{wt}}$ & -- & $0.0$ m & 10$\%$ +${\mathbf{x}_{\text{wt}}}$ & -- & $0.0$ m & 10$\%$ \\ -${{y_{slip}}^{max}}$ & -- & $30.0$ m & 10$\%$ +${{y_{\text{slip}}}^{\text{max}}}$ & -- & $30.0$ m & 10$\%$ \\ -${{y_{slip}}^{min}}$ & -- & $0.0$ m & 10$\%$ +${{y_{\text{slip}}}^{\text{min}}}$ & -- & $0.0$ m & 10$\%$ \\ -${\mathbf{y}_{slope}}$ & -- & $0.0$ m & 10$\%$ +${\mathbf{y}_{\text{slope}}}$ & -- & $0.0$ m & 10$\%$ \\ -${\mathbf{y}_{wt}}$ & -- & $0.0$ m & 10$\%$ +${\mathbf{y}_{\text{wt}}}$ & -- & $0.0$ m & 10$\%$ \\ -${γ_{dry}}$ & ${γ_{dry}}>0$ & $20.0\cdot{}10^{3}$ $\frac{\text{N}}{\text{m}^{3}}$ & 10$\%$ +${γ_{\text{dry}}}$ & ${γ_{\text{dry}}}>0$ & $20.0\cdot{}10^{3}$ $\frac{\text{N}}{\text{m}^{3}}$ & 10$\%$ \\ -${γ_{Sat}}$ & ${γ_{Sat}}>0$ & $20.0\cdot{}10^{3}$ $\frac{\text{N}}{\text{m}^{3}}$ & 10$\%$ +${γ_{\text{Sat}}}$ & ${γ_{\text{Sat}}}>0$ & $20.0\cdot{}10^{3}$ $\frac{\text{N}}{\text{m}^{3}}$ & 10$\%$ \\ ${γ_{w}}$ & ${γ_{w}}>0$ & $9.8\cdot{}10^{3}$ $\frac{\text{N}}{\text{m}^{3}}$ & 10$\%$ \\ @@ -2194,9 +2194,9 @@ \subsubsection{Properties of a Correct Solution} \\ \midrule \endhead -${F_{S}}$ & ${F_{S}}>0$ +${F_{\text{S}}}$ & ${F_{\text{S}}}>0$ \\ -$(x,y)$ & -- +$\text{(x,y)}$ & -- \\ \bottomrule \caption{Output Data Constraints} @@ -2226,35 +2226,35 @@ \subsection{Functional Requirements} \\ \midrule \endhead -$(x,y)$ & Cartesian position coordinates & m +$\text{(x,y)}$ & Cartesian position coordinates & m \\ $c'$ & Effective cohesion & Pa \\ $const_f$ & Decision on f & -- \\ -${{x_{slip}}^{maxEtr}}$ & Maximum entry x-coordinate & m +${{x_{\text{slip}}}^{\text{maxEtr}}}$ & Maximum entry x-coordinate & m \\ -${{x_{slip}}^{maxExt}}$ & Maximum exit x-coordinate & m +${{x_{\text{slip}}}^{\text{maxExt}}}$ & Maximum exit x-coordinate & m \\ -${{x_{slip}}^{minEtr}}$ & Minimum exit x-coordinate & m +${{x_{\text{slip}}}^{\text{minEtr}}}$ & Minimum exit x-coordinate & m \\ -${{x_{slip}}^{minExt}}$ & Minimum exit x-coordinate & m +${{x_{\text{slip}}}^{\text{minExt}}}$ & Minimum exit x-coordinate & m \\ -${\mathbf{x}_{slope}}$ & X-coordinates of the slope & m +${\mathbf{x}_{\text{slope}}}$ & X-coordinates of the slope & m \\ -${\mathbf{x}_{wt}}$ & X-coordinates of the water table & m +${\mathbf{x}_{\text{wt}}}$ & X-coordinates of the water table & m \\ -${{y_{slip}}^{max}}$ & Maximum y-coordinate & m +${{y_{\text{slip}}}^{\text{max}}}$ & Maximum y-coordinate & m \\ -${{y_{slip}}^{min}}$ & Minimum y-coordinate & m +${{y_{\text{slip}}}^{\text{min}}}$ & Minimum y-coordinate & m \\ -${\mathbf{y}_{slope}}$ & Y-coordinates of the slope & m +${\mathbf{y}_{\text{slope}}}$ & Y-coordinates of the slope & m \\ -${\mathbf{y}_{wt}}$ & Y-coordinates of the water table & m +${\mathbf{y}_{\text{wt}}}$ & Y-coordinates of the water table & m \\ -${γ_{dry}}$ & Soil dry unit weight & $\frac{\text{N}}{\text{m}^{3}}$ +${γ_{\text{dry}}}$ & Soil dry unit weight & $\frac{\text{N}}{\text{m}^{3}}$ \\ -${γ_{Sat}}$ & Soil saturated unit weight & $\frac{\text{N}}{\text{m}^{3}}$ +${γ_{\text{Sat}}}$ & Soil saturated unit weight & $\frac{\text{N}}{\text{m}^{3}}$ \\ ${γ_{w}}$ & Unit weight of water & $\frac{\text{N}}{\text{m}^{3}}$ \\ @@ -2272,17 +2272,17 @@ \subsection{Functional Requirements} \endhead $const_f$ & decision on f \\ -${{x_{slip}}^{maxExt}}$ & maximum exit x-coordinate +${{x_{\text{slip}}}^{\text{maxExt}}}$ & maximum exit x-coordinate \\ -${{x_{slip}}^{maxEtr}}$ & maximum entry x-coordinate +${{x_{\text{slip}}}^{\text{maxEtr}}}$ & maximum entry x-coordinate \\ -${{x_{slip}}^{minExt}}$ & minimum exit x-coordinate +${{x_{\text{slip}}}^{\text{minExt}}}$ & minimum exit x-coordinate \\ -${{x_{slip}}^{minEtr}}$ & minimum exit x-coordinate +${{x_{\text{slip}}}^{\text{minEtr}}}$ & minimum exit x-coordinate \\ -${{y_{slip}}^{max}}$ & maximum y-coordinate +${{y_{\text{slip}}}^{\text{max}}}$ & maximum y-coordinate \\ -${{y_{slip}}^{min}}$ & minimum y-coordinate +${{y_{\text{slip}}}^{\text{min}}}$ & minimum y-coordinate \\ \bottomrule \caption{Inputs to be returned as output} diff --git a/code/stable/swhs/SRS/SWHS_SRS.tex b/code/stable/swhs/SRS/SWHS_SRS.tex index 791ac2c0d3..9f237afe23 100644 --- a/code/stable/swhs/SRS/SWHS_SRS.tex +++ b/code/stable/swhs/SRS/SWHS_SRS.tex @@ -66,9 +66,9 @@ \subsection{Table of Symbols} \endhead ${A_{C}}$ & Heating coil surface area & $\text{m}^{2}$ \\ -${A_{in}}$ & Surface area over which heat is transferred in & $\text{m}^{2}$ +${A_{\text{in}}}$ & Surface area over which heat is transferred in & $\text{m}^{2}$ \\ -${A_{out}}$ & Surface area over which heat is transferred out & $\text{m}^{2}$ +${A_{\text{out}}}$ & Surface area over which heat is transferred out & $\text{m}^{2}$ \\ ${A_{P}}$ & Phase change material surface area & $\text{m}^{2}$ \\ @@ -84,7 +84,7 @@ \subsection{Table of Symbols} \\ ${{C_{P}}^{S}}$ & Specific heat capacity of PCM as a solid & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ \\ -${C_{tol}}$ & Relative tolerance for conservation of energy & -- +${C_{\text{tol}}}$ & Relative tolerance for conservation of energy & -- \\ ${C_{W}}$ & Specific heat capacity of water & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ \\ @@ -96,7 +96,7 @@ \subsection{Table of Symbols} \\ ${E_{W}}$ & Change in heat energy in the water & J \\ -${{{E_{P}}_{melt}}^{init}}$ & Change in heat energy in the PCM at the instant when melting begins & J +${{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}$ & Change in heat energy in the PCM at the instant when melting begins & J \\ $g$ & Volumetric heat generation per unit volume & $\frac{\text{W}}{\text{m}^{3}}$ \\ @@ -106,7 +106,7 @@ \subsection{Table of Symbols} \\ ${h_{C}}$ & Convective heat transfer coefficient between coil and water & $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ \\ -${h_{min}}$ & Minimum thickness of a sheet of PCM & m +${h_{\text{min}}}$ & Minimum thickness of a sheet of PCM & m \\ ${h_{P}}$ & Convective heat transfer coefficient between PCM and water & $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ \\ @@ -128,9 +128,9 @@ \subsection{Table of Symbols} \\ ${q_{C}}$ & Heat flux into the water from the coil & $\frac{\text{W}}{\text{m}^{2}}$ \\ -${q_{in}}$ & Heat flux input & $\frac{\text{W}}{\text{m}^{2}}$ +${q_{\text{in}}}$ & Heat flux input & $\frac{\text{W}}{\text{m}^{2}}$ \\ -${q_{out}}$ & Heat flux output & $\frac{\text{W}}{\text{m}^{2}}$ +${q_{\text{out}}}$ & Heat flux output & $\frac{\text{W}}{\text{m}^{2}}$ \\ ${q_{P}}$ & Heat flux into the PCM from water & $\frac{\text{W}}{\text{m}^{2}}$ \\ @@ -142,17 +142,17 @@ \subsection{Table of Symbols} \\ $ΔT$ & Change in temperature & ${}^{\circ}$C \\ -${T_{boil}}$ & Boiling point temperature & ${}^{\circ}$C +${T_{\text{boil}}}$ & Boiling point temperature & ${}^{\circ}$C \\ ${T_{C}}$ & Temperature of the heating coil & ${}^{\circ}$C \\ -${T_{env}}$ & Temperature of the environment & ${}^{\circ}$C +${T_{\text{env}}}$ & Temperature of the environment & ${}^{\circ}$C \\ -${T_{init}}$ & Initial temperature & ${}^{\circ}$C +${T_{\text{init}}}$ & Initial temperature & ${}^{\circ}$C \\ -${T_{melt}}$ & Melting point temperature & ${}^{\circ}$C +${T_{\text{melt}}}$ & Melting point temperature & ${}^{\circ}$C \\ -${{T_{melt}}^{P}}$ & Melting point temperature for PCM & ${}^{\circ}$C +${{T_{\text{melt}}}^{P}}$ & Melting point temperature for PCM & ${}^{\circ}$C \\ ${T_{P}}$ & Temperature of the phase change material & ${}^{\circ}$C \\ @@ -160,19 +160,19 @@ \subsection{Table of Symbols} \\ $t$ & Time & s \\ -${t_{final}}$ & Final time & s +${t_{\text{final}}}$ & Final time & s \\ -${{t_{melt}}^{final}}$ & Time at which melting of PCM ends & s +${{t_{\text{melt}}}^{\text{final}}}$ & Time at which melting of PCM ends & s \\ -${{t_{melt}}^{init}}$ & Time at which melting of PCM begins & s +${{t_{\text{melt}}}^{\text{init}}}$ & Time at which melting of PCM begins & s \\ -${t_{step}}$ & Time step for simulation & s +${t_{\text{step}}}$ & Time step for simulation & s \\ $V$ & Volume & $\text{m}^{3}$ \\ ${V_{P}}$ & Volume of PCM & $\text{m}^{3}$ \\ -${V_{tank}}$ & Volume of the cylindrical tank & $\text{m}^{3}$ +${V_{\text{tank}}}$ & Volume of the cylindrical tank & $\text{m}^{3}$ \\ ${V_{W}}$ & Volume of water & $\text{m}^{3}$ \\ @@ -412,9 +412,9 @@ \subsubsection{Theoretical Models} \\ \midrule \\ Equation & \begin{displaymath} E=\begin{cases} - {C^{S}} m ΔT, & T<{T_{melt}}\\ -{C^{L}} m ΔT, & {T_{melt}}{{T_{melt}}^{P}}\\ -0, & {T_{P}}={{T_{melt}}^{P}}\\ + \frac{1}{{{τ_{P}}^{S}}} \left({T_{W}}\left(t\right)-{T_{P}}\left(t\right)\right), & {T_{P}}<{{T_{\text{melt}}}^{P}}\\ +\frac{1}{{{τ_{P}}^{L}}} \left({T_{W}}\left(t\right)-{T_{P}}\left(t\right)\right), & {T_{P}}>{{T_{\text{melt}}}^{P}}\\ +0, & {T_{P}}={{T_{\text{melt}}}^{P}}\\ 0, & 0<ϕ<1 \end{cases} \end{displaymath} @@ -871,11 +871,11 @@ \subsubsection{Instance Models} \item{${{τ_{P}}^{S}}$ is the ODE parameter for solid PCM (s)} \item{${T_{W}}$ is the temperature of the water (${}^{\circ}$C)} \item{${{τ_{P}}^{L}}$ is the ODE parameter for liquid PCM (s)} - \item{${{T_{melt}}^{P}}$ is the melting point temperature for PCM (${}^{\circ}$C)} + \item{${{T_{\text{melt}}}^{P}}$ is the melting point temperature for PCM (${}^{\circ}$C)} \item{$ϕ$ is the melt fraction (Unitless)} \end{symbDescription} \\ \midrule \\ -Notes & ${{T_{melt}}^{P}}$, ${t_{final}}$, ${T_{init}}$, ${h_{P}}$, ${m_{P}}$, ${{C_{P}}^{S}}$, ${{C_{P}}^{S}}$ from (\hyperref[IM:eBalanceOnWtr]{IM: eBalanceOnWtr}). The input is constrained so that ${T_{init}}<{{T_{melt}}^{P}}$ (\hyperref[assumpPIS]{A: PCM-Initially-Solid}) ${T_{P}}$, $0{{T_{melt}}^{P}}\\ -{{{E_{P}}_{melt}}^{init}}+{Q_{P}}\left(t\right), & {T_{P}}={{T_{melt}}^{P}}\\ -{{{E_{P}}_{melt}}^{init}}+{Q_{P}}\left(t\right), & 0<ϕ<1 + {{C_{P}}^{S}} {m_{P}} \left({T_{P}}\left(t\right)-{T_{\text{init}}}\right), & {T_{P}}<{{T_{\text{melt}}}^{P}}\\ +{{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}+{H_{f}} {m_{P}}+{{C_{P}}^{L}} {m_{P}} \left({T_{P}}\left(t\right)-{{T_{\text{melt}}}^{P}}\right), & {T_{P}}>{{T_{\text{melt}}}^{P}}\\ +{{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}+{Q_{P}}\left(t\right), & {T_{P}}={{T_{\text{melt}}}^{P}}\\ +{{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}+{Q_{P}}\left(t\right), & 0<ϕ<1 \end{cases} \end{displaymath} \\ \midrule \\ @@ -982,16 +982,16 @@ \subsubsection{Instance Models} \item{${m_{P}}$ is the mass of phase change material (kg)} \item{${T_{P}}$ is the temperature of the phase change material (${}^{\circ}$C)} \item{$t$ is the time (s)} - \item{${T_{init}}$ is the initial temperature (${}^{\circ}$C)} - \item{${{{E_{P}}_{melt}}^{init}}$ is the change in heat energy in the PCM at the instant when melting begins (J)} + \item{${T_{\text{init}}}$ is the initial temperature (${}^{\circ}$C)} + \item{${{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}$ is the change in heat energy in the PCM at the instant when melting begins (J)} \item{${H_{f}}$ is the specific latent heat of fusion ($\frac{\text{J}}{\text{kg}}$)} \item{${{C_{P}}^{L}}$ is the specific heat capacity of PCM as a liquid ($\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$)} - \item{${{T_{melt}}^{P}}$ is the melting point temperature for PCM (${}^{\circ}$C)} + \item{${{T_{\text{melt}}}^{P}}$ is the melting point temperature for PCM (${}^{\circ}$C)} \item{${Q_{P}}$ is the latent heat energy added to PCM (J)} \item{$ϕ$ is the melt fraction (Unitless)} \end{symbDescription} \\ \midrule \\ -Notes & The above equation is derived using \hyperref[TM:sensHtE]{TM: sensHtE} and \hyperref[TM:latentHtE]{TM: latentHtE}. ${E_{P}}$ is the change in thermal energy of the PCM relative to the energy at the initial temperature (${T_{init}}$) J. ${E_{P}}$ for the solid PCM is found using \hyperref[TM:sensHtE]{TM: sensHtE} for sensible heating, with the specific heat capacity of the solid PCM, ${{C_{P}}^{S}}$ ($\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$) and the change in the PCM temperature from the initial temperature (${}^{\circ}$C). ${E_{P}}$ for the melted PCM (${T_{P}}>{{{E_{P}}_{melt}}^{init}}$) is found using \hyperref[TM:sensHtE]{TM: sensHtE} for sensible heat of the liquid. PCM plus the energy when melting starts, plus the energy required to melt all of the PCM. The energy when melting starts is ${{{E_{P}}_{melt}}^{init}}$ (J). The energy required to melt all of the PCM is ${H_{f}} {m_{P}}$ (J) (\hyperref[DD:htFusion]{DD: htFusion}). The specific heat capacity of the liquid PCM is ${{C_{P}}^{L}}$ ($\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$) and the change in temperature is ${T_{P}}-{{T_{melt}}^{P}}$ (${}^{\circ}$C). ${E_{P}}$ during melting of the PCM is found using the energy required at the instant melting of the PCM begins, ${{{E_{P}}_{melt}}^{init}}$ plus the latent heat energy added to the PCM, ${Q_{P}}$ (J) since the time when melting began ${{t_{melt}}^{init}}$ (s). The heat energy for boiling of the PCM is not detailed, since the PCM is assumed to either be in a solid or liquid state (\hyperref[assumpNGSP]{A: No-Gaseous-State-PCM}) (\hyperref[assumpPIS]{A: PCM-Initially-Solid}). +Notes & The above equation is derived using \hyperref[TM:sensHtE]{TM: sensHtE} and \hyperref[TM:latentHtE]{TM: latentHtE}. ${E_{P}}$ is the change in thermal energy of the PCM relative to the energy at the initial temperature (${T_{\text{init}}}$) J. ${E_{P}}$ for the solid PCM is found using \hyperref[TM:sensHtE]{TM: sensHtE} for sensible heating, with the specific heat capacity of the solid PCM, ${{C_{P}}^{S}}$ ($\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$) and the change in the PCM temperature from the initial temperature (${}^{\circ}$C). ${E_{P}}$ for the melted PCM (${T_{P}}>{{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}$) is found using \hyperref[TM:sensHtE]{TM: sensHtE} for sensible heat of the liquid. PCM plus the energy when melting starts, plus the energy required to melt all of the PCM. The energy when melting starts is ${{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}$ (J). The energy required to melt all of the PCM is ${H_{f}} {m_{P}}$ (J) (\hyperref[DD:htFusion]{DD: htFusion}). The specific heat capacity of the liquid PCM is ${{C_{P}}^{L}}$ ($\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$) and the change in temperature is ${T_{P}}-{{T_{\text{melt}}}^{P}}$ (${}^{\circ}$C). ${E_{P}}$ during melting of the PCM is found using the energy required at the instant melting of the PCM begins, ${{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}$ plus the latent heat energy added to the PCM, ${Q_{P}}$ (J) since the time when melting began ${{t_{\text{melt}}}^{\text{init}}}$ (s). The heat energy for boiling of the PCM is not detailed, since the PCM is assumed to either be in a solid or liquid state (\hyperref[assumpNGSP]{A: No-Gaseous-State-PCM}) (\hyperref[assumpPIS]{A: PCM-Initially-Solid}). \\ \midrule \\ Source & \cite{koothoor2013} \\ \midrule \\ @@ -1001,48 +1001,48 @@ \subsubsection{Instance Models} \end{minipage} \subsubsection{Data Constraints} \label{Sec:DataConstraints} -\hyperref[Table:InDataConstraints]{Table:InDataConstraints} shows the data constraints on the input variables. The column for physical constraints gives the physical limitations on the range of values that can be taken by the variable. The uncertainty column provides an estimate of the confidence with which the physical quantities can be measured. This information would be part of the input if one were performing an uncertainty quantification exercise. The constraints are conservative, to give the user of the model the flexibility to experiment with unusual situations. The column of typical values is intended to provide a feel for a common scenario. The column for software constraints restricts the range of inputs to reasonable values.(*) These quantities cannot be equal to zero, or there will be a divide by zero in the model. (+) These quantities cannot be zero, or there would be freezing (\hyperref[assumpPIS]{A: PCM-Initially-Solid}). (++) The constraints on the surface area are calculated by considering the surface area to volume ratio. The assumption is that the lowest ratio is 1 and the highest possible is $\frac{2}{{h_{min}}}$, where ${h_{min}}$ is the thickness of a ``sheet'' of PCM. A thin sheet has the greatest surface area to volume ratio. (**) The constraint on the maximum time at the end of the simulation is the total number of seconds in one day. +\hyperref[Table:InDataConstraints]{Table:InDataConstraints} shows the data constraints on the input variables. The column for physical constraints gives the physical limitations on the range of values that can be taken by the variable. The uncertainty column provides an estimate of the confidence with which the physical quantities can be measured. This information would be part of the input if one were performing an uncertainty quantification exercise. The constraints are conservative, to give the user of the model the flexibility to experiment with unusual situations. The column of typical values is intended to provide a feel for a common scenario. The column for software constraints restricts the range of inputs to reasonable values.(*) These quantities cannot be equal to zero, or there will be a divide by zero in the model. (+) These quantities cannot be zero, or there would be freezing (\hyperref[assumpPIS]{A: PCM-Initially-Solid}). (++) The constraints on the surface area are calculated by considering the surface area to volume ratio. The assumption is that the lowest ratio is 1 and the highest possible is $\frac{2}{{h_{\text{min}}}}$, where ${h_{\text{min}}}$ is the thickness of a ``sheet'' of PCM. A thin sheet has the greatest surface area to volume ratio. (**) The constraint on the maximum time at the end of the simulation is the total number of seconds in one day. \begin{longtable}{l l l l l} \toprule \textbf{Var} & \textbf{Physical Constraints} & \textbf{Software Constraints} & \textbf{Typical Value} & \textbf{Uncert.} \\ \midrule \endhead -${A_{C}}$ & ${A_{C}}>0$ & ${A_{C}}\leq{}{{A_{C}}^{max}}$ & $0.12$ $\text{m}^{2}$ & 10$\%$ +${A_{C}}$ & ${A_{C}}>0$ & ${A_{C}}\leq{}{{A_{C}}^{\text{max}}}$ & $0.12$ $\text{m}^{2}$ & 10$\%$ \\ -${A_{P}}$ & ${A_{P}}>0$ & ${V_{P}}\leq{}{A_{P}}\leq{}\frac{2}{{h_{min}}} {V_{tank}}$ & $1.2$ $\text{m}^{2}$ & 10$\%$ +${A_{P}}$ & ${A_{P}}>0$ & ${V_{P}}\leq{}{A_{P}}\leq{}\frac{2}{{h_{\text{min}}}} {V_{\text{tank}}}$ & $1.2$ $\text{m}^{2}$ & 10$\%$ \\ -${{C_{P}}^{L}}$ & ${{C_{P}}^{L}}>0$ & ${{{C_{P}}^{L}}_{min}}<{{C_{P}}^{L}}<{{{C_{P}}^{L}}_{max}}$ & $2.27\cdot{}10^{3}$ $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ & 10$\%$ +${{C_{P}}^{L}}$ & ${{C_{P}}^{L}}>0$ & ${{{C_{P}}^{L}}_{\text{min}}}<{{C_{P}}^{L}}<{{{C_{P}}^{L}}_{\text{max}}}$ & $2.27\cdot{}10^{3}$ $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ & 10$\%$ \\ -${{C_{P}}^{S}}$ & ${{C_{P}}^{S}}>0$ & ${{{C_{P}}^{S}}_{min}}<{{C_{P}}^{S}}<{{{C_{P}}^{S}}_{max}}$ & $1.76\cdot{}10^{3}$ $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ & 10$\%$ +${{C_{P}}^{S}}$ & ${{C_{P}}^{S}}>0$ & ${{{C_{P}}^{S}}_{\text{min}}}<{{C_{P}}^{S}}<{{{C_{P}}^{S}}_{\text{max}}}$ & $1.76\cdot{}10^{3}$ $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ & 10$\%$ \\ -${C_{W}}$ & ${C_{W}}>0$ & ${{C_{W}}^{min}}<{C_{W}}<{{C_{W}}^{max}}$ & $4.186\cdot{}10^{3}$ $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ & 10$\%$ +${C_{W}}$ & ${C_{W}}>0$ & ${{C_{W}}^{\text{min}}}<{C_{W}}<{{C_{W}}^{\text{max}}}$ & $4.186\cdot{}10^{3}$ $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ & 10$\%$ \\ $D$ & $D>0$ & -- & $0.412$ m & 10$\%$ \\ -${H_{f}}$ & ${H_{f}}>0$ & ${{H_{f}}_{min}}<{H_{f}}<{{H_{f}}_{max}}$ & $211.6\cdot{}10^{3}$ $\frac{\text{J}}{\text{kg}}$ & 10$\%$ +${H_{f}}$ & ${H_{f}}>0$ & ${{H_{f}}_{\text{min}}}<{H_{f}}<{{H_{f}}_{\text{max}}}$ & $211.6\cdot{}10^{3}$ $\frac{\text{J}}{\text{kg}}$ & 10$\%$ \\ -${h_{C}}$ & ${h_{C}}>0$ & ${{h_{C}}^{min}}\leq{}{h_{C}}\leq{}{{h_{C}}^{max}}$ & $1.0\cdot{}10^{3}$ $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ & 10$\%$ +${h_{C}}$ & ${h_{C}}>0$ & ${{h_{C}}^{\text{min}}}\leq{}{h_{C}}\leq{}{{h_{C}}^{\text{max}}}$ & $1.0\cdot{}10^{3}$ $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ & 10$\%$ \\ -${h_{P}}$ & ${h_{P}}>0$ & ${{h_{P}}^{min}}\leq{}{h_{P}}\leq{}{{h_{P}}^{max}}$ & $1.0\cdot{}10^{3}$ $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ & 10$\%$ +${h_{P}}$ & ${h_{P}}>0$ & ${{h_{P}}^{\text{min}}}\leq{}{h_{P}}\leq{}{{h_{P}}^{\text{max}}}$ & $1.0\cdot{}10^{3}$ $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ & 10$\%$ \\ -$L$ & $L>0$ & ${L_{min}}\leq{}L\leq{}{L_{max}}$ & $1.5$ m & 10$\%$ +$L$ & $L>0$ & ${L_{\text{min}}}\leq{}L\leq{}{L_{\text{max}}}$ & $1.5$ m & 10$\%$ \\ ${T_{C}}$ & $0<{T_{C}}<100$ & -- & $50.0$ ${}^{\circ}$C & 10$\%$ \\ -${T_{init}}$ & $0<{T_{init}}<{T_{melt}}$ & -- & $40.0$ ${}^{\circ}$C & 10$\%$ +${T_{\text{init}}}$ & $0<{T_{\text{init}}}<{T_{\text{melt}}}$ & -- & $40.0$ ${}^{\circ}$C & 10$\%$ \\ -${{T_{melt}}^{P}}$ & $0<{{T_{melt}}^{P}}<{T_{C}}$ & -- & $44.2$ ${}^{\circ}$C & 10$\%$ +${{T_{\text{melt}}}^{P}}$ & $0<{{T_{\text{melt}}}^{P}}<{T_{C}}$ & -- & $44.2$ ${}^{\circ}$C & 10$\%$ \\ -${t_{final}}$ & ${t_{final}}>0$ & ${t_{final}}<{{t_{final}}^{max}}$ & $50.0\cdot{}10^{3}$ s & 10$\%$ +${t_{\text{final}}}$ & ${t_{\text{final}}}>0$ & ${t_{\text{final}}}<{{t_{\text{final}}}^{\text{max}}}$ & $50.0\cdot{}10^{3}$ s & 10$\%$ \\ -${t_{step}}$ & $0<{t_{step}}<{t_{final}}$ & -- & $0.01$ s & 10$\%$ +${t_{\text{step}}}$ & $0<{t_{\text{step}}}<{t_{\text{final}}}$ & -- & $0.01$ s & 10$\%$ \\ -${V_{P}}$ & $0<{V_{P}}<{V_{tank}}$ & ${V_{P}}\geq{}MINFRACT {V_{tank}}$ & $0.05$ $\text{m}^{3}$ & 10$\%$ +${V_{P}}$ & $0<{V_{P}}<{V_{\text{tank}}}$ & ${V_{P}}\geq{}MINFRACT {V_{\text{tank}}}$ & $0.05$ $\text{m}^{3}$ & 10$\%$ \\ -${ρ_{P}}$ & ${ρ_{P}}>0$ & ${{ρ_{P}}^{min}}<{ρ_{P}}<{{ρ_{P}}^{max}}$ & $1.007\cdot{}10^{3}$ $\frac{\text{kg}}{\text{m}^{3}}$ & 10$\%$ +${ρ_{P}}$ & ${ρ_{P}}>0$ & ${{ρ_{P}}^{\text{min}}}<{ρ_{P}}<{{ρ_{P}}^{\text{max}}}$ & $1.007\cdot{}10^{3}$ $\frac{\text{kg}}{\text{m}^{3}}$ & 10$\%$ \\ -${ρ_{W}}$ & ${ρ_{W}}>0$ & ${{ρ_{W}}^{min}}<{ρ_{W}}\leq{}{{ρ_{W}}^{max}}$ & $1.0\cdot{}10^{3}$ $\frac{\text{kg}}{\text{m}^{3}}$ & 10$\%$ +${ρ_{W}}$ & ${ρ_{W}}>0$ & ${{ρ_{W}}^{\text{min}}}<{ρ_{W}}\leq{}{{ρ_{W}}^{\text{max}}}$ & $1.0\cdot{}10^{3}$ $\frac{\text{kg}}{\text{m}^{3}}$ & 10$\%$ \\ \bottomrule \caption{Input Data Constraints} @@ -1057,9 +1057,9 @@ \subsubsection{Properties of a Correct Solution} \\ \midrule \endhead -${T_{W}}$ & ${T_{init}}\leq{}{T_{W}}\leq{}{T_{C}}$ +${T_{W}}$ & ${T_{\text{init}}}\leq{}{T_{W}}\leq{}{T_{C}}$ \\ -${T_{P}}$ & ${T_{init}}\leq{}{T_{P}}\leq{}{T_{C}}$ +${T_{P}}$ & ${T_{\text{init}}}\leq{}{T_{P}}\leq{}{T_{C}}$ \\ ${E_{W}}$ & ${E_{W}}\geq{}0$ \\ @@ -1077,7 +1077,7 @@ \subsubsection{Properties of a Correct Solution} \begin{displaymath} {E_{P}}=\int_{0}^{t}{{h_{P}} {A_{P}} \left({T_{W}}\left(t\right)-{T_{P}}\left(t\right)\right)}\,dt \end{displaymath} -Equations (FIXME: Equation 7) and (FIXME: Equation 8) can be used as ``sanity'' checks to gain confidence in any solution computed by SWHS. The relative error between the results computed by SWHS and the results calculated from the RHS of these equations should be less than ${C_{tol}}$ \hyperref[verifyEnergyOutput]{FR: Verify-Energy-Output-Follow-Conservation-of-Energy}. +Equations (FIXME: Equation 7) and (FIXME: Equation 8) can be used as ``sanity'' checks to gain confidence in any solution computed by SWHS. The relative error between the results computed by SWHS and the results calculated from the RHS of these equations should be less than ${C_{\text{tol}}}$ \hyperref[verifyEnergyOutput]{FR: Verify-Energy-Output-Follow-Conservation-of-Energy}. \section{Requirements} \label{Sec:Requirements} This section provides the functional requirements, the tasks and behaviours that the software is expected to complete, and the non-functional requirements, the qualities that the software is expected to exhibit. @@ -1086,16 +1086,16 @@ \subsection{Functional Requirements} This section provides the functional requirements, the tasks and behaviours that the software is expected to complete. \begin{itemize} \item[Input-Initial-Quantities:\phantomsection\label{inputInitQuants}]Input the following quantities described in \hyperref[Table:ReqInputs]{Table:ReqInputs}, which define the tank parameters, material properties and initial conditions. -\item[Find-Mass:\phantomsection\label{findMass}]Use the inputs in \hyperref[inputInitQuants]{FR: Input-Initial-Quantities} to find the masses needed for \hyperref[IM:eBalanceOnWtr]{IM: eBalanceOnWtr}, \hyperref[IM:eBalanceOnPCM]{IM: eBalanceOnPCM}, \hyperref[IM:heatEInWtr]{IM: heatEInWtr}, and \hyperref[IM:heatEInPCM]{IM: heatEInPCM}, using ${m_{W}}={V_{W}} {ρ_{W}}=\left({V_{tank}}-{V_{P}}\right) {ρ_{W}}=\left(π \left(\frac{D}{2}\right)^{2} L-{V_{P}}\right) {ρ_{W}}$, ${m_{P}}={V_{P}} {ρ_{P}}$, and \hyperref[assumpVCN]{A: Volume-Coil-Negligible}, where ${V_{W}}$ is the volume of water and ${V_{tank}}$ is the volume of the cylindrical tank. +\item[Find-Mass:\phantomsection\label{findMass}]Use the inputs in \hyperref[inputInitQuants]{FR: Input-Initial-Quantities} to find the masses needed for \hyperref[IM:eBalanceOnWtr]{IM: eBalanceOnWtr}, \hyperref[IM:eBalanceOnPCM]{IM: eBalanceOnPCM}, \hyperref[IM:heatEInWtr]{IM: heatEInWtr}, and \hyperref[IM:heatEInPCM]{IM: heatEInPCM}, using ${m_{W}}={V_{W}} {ρ_{W}}=\left({V_{\text{tank}}}-{V_{P}}\right) {ρ_{W}}=\left(π \left(\frac{D}{2}\right)^{2} L-{V_{P}}\right) {ρ_{W}}$, ${m_{P}}={V_{P}} {ρ_{P}}$, and \hyperref[assumpVCN]{A: Volume-Coil-Negligible}, where ${V_{W}}$ is the volume of water and ${V_{\text{tank}}}$ is the volume of the cylindrical tank. \item[Check-Input-with-Physical\_Constraints:\phantomsection\label{checkWithPhysConsts}]Verify that the inputs satisfy the required physical constraints shown in \hyperref[Sec:DataConstraints]{Section: Data Constraints}. \item[Output-Input-Derived-Quantities:\phantomsection\label{outputInputDerivQuants}]Output the input quantities and derived quantities in the following list: the quantities from \hyperref[inputInitQuants]{FR: Input-Initial-Quantities}, the masses from \hyperref[findMass]{FR: Find-Mass}, ${τ_{W}}$ (from \hyperref[IM:eBalanceOnWtr]{IM: eBalanceOnWtr}), $η$ (from \hyperref[IM:eBalanceOnWtr]{IM: eBalanceOnWtr}), ${{τ_{P}}^{S}}$ (from \hyperref[IM:eBalanceOnPCM]{IM: eBalanceOnPCM}), and ${{τ_{P}}^{L}}$ (from \hyperref[IM:eBalanceOnPCM]{IM: eBalanceOnPCM}). \item[Calculate-Temperature-Water-Over-Time:\phantomsection\label{calcTempWtrOverTime}]Calculate and output the temperature of the water (${T_{W}}$($t$)) over the simulation time (from \hyperref[IM:eBalanceOnWtr]{IM: eBalanceOnWtr}). \item[Calculate-Temperature-PCM-Over-Time:\phantomsection\label{calcTempPCMOverTime}]Calculate and output the temperature of the phase change material (${T_{P}}$($t$)) over the simulation time (from \hyperref[IM:eBalanceOnPCM]{IM: eBalanceOnPCM}). \item[Calculate-Change-Heat\_Energy-Water-Over-Time:\phantomsection\label{calcChgHeatEnergyWtrOverTime}]Calculate and output the change in heat energy in the water (${E_{W}}$($t$)) over the simulation time (from \hyperref[IM:heatEInWtr]{IM: heatEInWtr}). \item[Calculate-Change-Heat\_Energy-PCM-Over-Time:\phantomsection\label{calcChgHeatEnergyPCMOverTime}]Calculate and output the change in heat energy in the PCM (${E_{P}}$($t$)) over the simulation time (from \hyperref[IM:heatEInPCM]{IM: heatEInPCM}). -\item[Verify-Energy-Output-Follow-Conservation-of-Energy:\phantomsection\label{verifyEnergyOutput}]Verify that the energy outputs (${E_{W}}$($t$) and ${E_{P}}$($t$)) follow the law of conservation of energy, as outlined in \hyperref[Sec:CorSolProps]{Section: Properties of a Correct Solution}, with relative error no greater than ${C_{tol}}$. -\item[Calculate-PCM-Melt-Begin-Time:\phantomsection\label{calcPCMMeltBegin}]Calculate and output the time at which the PCM begins to melt ${{t_{melt}}^{init}}$ (from \hyperref[IM:eBalanceOnPCM]{IM: eBalanceOnPCM}). -\item[Calculate-PCM-Melt-End-Time:\phantomsection\label{calcPCMMeltEnd}]Calculate and output the time at which the PCM stops melting ${{t_{melt}}^{final}}$ (from \hyperref[IM:eBalanceOnPCM]{IM: eBalanceOnPCM}). +\item[Verify-Energy-Output-Follow-Conservation-of-Energy:\phantomsection\label{verifyEnergyOutput}]Verify that the energy outputs (${E_{W}}$($t$) and ${E_{P}}$($t$)) follow the law of conservation of energy, as outlined in \hyperref[Sec:CorSolProps]{Section: Properties of a Correct Solution}, with relative error no greater than ${C_{\text{tol}}}$. +\item[Calculate-PCM-Melt-Begin-Time:\phantomsection\label{calcPCMMeltBegin}]Calculate and output the time at which the PCM begins to melt ${{t_{\text{melt}}}^{\text{init}}}$ (from \hyperref[IM:eBalanceOnPCM]{IM: eBalanceOnPCM}). +\item[Calculate-PCM-Melt-End-Time:\phantomsection\label{calcPCMMeltEnd}]Calculate and output the time at which the PCM stops melting ${{t_{\text{melt}}}^{\text{final}}}$ (from \hyperref[IM:eBalanceOnPCM]{IM: eBalanceOnPCM}). \end{itemize} \begin{longtabu}{l X[l] l} \toprule @@ -1107,7 +1107,7 @@ \subsection{Functional Requirements} \\ ${A_{P}}$ & Phase change material surface area & $\text{m}^{2}$ \\ -${A_{tol}}$ & Absolute tolerance & -- +${A_{\text{tol}}}$ & Absolute tolerance & -- \\ ${{C_{P}}^{L}}$ & Specific heat capacity of PCM as a liquid & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ \\ @@ -1125,17 +1125,17 @@ \subsection{Functional Requirements} \\ $L$ & Length of tank & m \\ -${R_{tol}}$ & Relative tolerance & -- +${R_{\text{tol}}}$ & Relative tolerance & -- \\ ${T_{C}}$ & Temperature of the heating coil & ${}^{\circ}$C \\ -${T_{init}}$ & Initial temperature & ${}^{\circ}$C +${T_{\text{init}}}$ & Initial temperature & ${}^{\circ}$C \\ -${{T_{melt}}^{P}}$ & Melting point temperature for PCM & ${}^{\circ}$C +${{T_{\text{melt}}}^{P}}$ & Melting point temperature for PCM & ${}^{\circ}$C \\ -${t_{final}}$ & Final time & s +${t_{\text{final}}}$ & Final time & s \\ -${t_{step}}$ & Time step for simulation & s +${t_{\text{step}}}$ & Time step for simulation & s \\ ${V_{P}}$ & Volume of PCM & $\text{m}^{3}$ \\ @@ -1368,49 +1368,49 @@ \section{Values of Auxiliary Constants} \\ \midrule \endhead -${{A_{C}}^{max}}$ & maximum surface area of coil & $100000$ & $\text{m}^{2}$ +${{A_{C}}^{\text{max}}}$ & maximum surface area of coil & $100000$ & $\text{m}^{2}$ \\ -${C_{tol}}$ & relative tolerance for conservation of energy & $0.001\%$ & -- +${C_{\text{tol}}}$ & relative tolerance for conservation of energy & $0.001\%$ & -- \\ -${{C_{W}}^{max}}$ & maximum specific heat capacity of water & $4210$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ +${{C_{W}}^{\text{max}}}$ & maximum specific heat capacity of water & $4210$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ \\ -${{C_{W}}^{min}}$ & minimum specific heat capacity of water & $4170$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ +${{C_{W}}^{\text{min}}}$ & minimum specific heat capacity of water & $4170$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ \\ -${{{C_{P}}^{L}}_{max}}$ & maximum specific heat capacity of PCM as a liquid & $5000$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ +${{{C_{P}}^{L}}_{\text{max}}}$ & maximum specific heat capacity of PCM as a liquid & $5000$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ \\ -${{{C_{P}}^{L}}_{min}}$ & minimum specific heat capacity of PCM as a liquid & $100$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ +${{{C_{P}}^{L}}_{\text{min}}}$ & minimum specific heat capacity of PCM as a liquid & $100$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ \\ -${{{C_{P}}^{S}}_{max}}$ & maximum specific heat capacity of PCM as a solid & $4000$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ +${{{C_{P}}^{S}}_{\text{max}}}$ & maximum specific heat capacity of PCM as a solid & $4000$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ \\ -${{{C_{P}}^{S}}_{min}}$ & minimum specific heat capacity of PCM as a solid & $100$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ +${{{C_{P}}^{S}}_{\text{min}}}$ & minimum specific heat capacity of PCM as a solid & $100$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ \\ -${{H_{f}}_{max}}$ & maximum specific latent heat of fusion & $1000000$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ +${{H_{f}}_{\text{max}}}$ & maximum specific latent heat of fusion & $1000000$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ \\ -${{H_{f}}_{min}}$ & minimum specific latent heat of fusion & $0$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ +${{H_{f}}_{\text{min}}}$ & minimum specific latent heat of fusion & $0$ & $\frac{\text{J}}{(\text{kg}{}^{\circ}\text{C})}$ \\ -${{h_{C}}^{max}}$ & maximum convective heat transfer coefficient between coil and water & $10000$ & $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ +${{h_{C}}^{\text{max}}}$ & maximum convective heat transfer coefficient between coil and water & $10000$ & $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ \\ -${{h_{C}}^{min}}$ & minimum convective heat transfer coefficient between coil and water & $10$ & $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ +${{h_{C}}^{\text{min}}}$ & minimum convective heat transfer coefficient between coil and water & $10$ & $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ \\ -${{h_{P}}^{max}}$ & maximum convective heat transfer coefficient between PCM and water & $10000$ & $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ +${{h_{P}}^{\text{max}}}$ & maximum convective heat transfer coefficient between PCM and water & $10000$ & $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ \\ -${{h_{P}}^{min}}$ & minimum convective heat transfer coefficient between PCM and water & $10$ & $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ +${{h_{P}}^{\text{min}}}$ & minimum convective heat transfer coefficient between PCM and water & $10$ & $\frac{\text{W}}{(\text{m}^{2}{}^{\circ}\text{C})}$ \\ -${L_{max}}$ & maximum length of tank & $50$ & m +${L_{\text{max}}}$ & maximum length of tank & $50$ & m \\ -${L_{min}}$ & minimum length of tank & $0.1$ & m +${L_{\text{min}}}$ & minimum length of tank & $0.1$ & m \\ $MINFRACT$ & minimum fraction of the tank volume taken up by the PCM & $1.0\cdot{}10^{-6}$ & -- \\ -${{t_{final}}^{max}}$ & maximum final time & $86400$ & s +${{t_{\text{final}}}^{\text{max}}}$ & maximum final time & $86400$ & s \\ -${{ρ_{P}}^{max}}$ & maximum density of PCM & $20000$ & $\frac{\text{kg}}{\text{m}^{3}}$ +${{ρ_{P}}^{\text{max}}}$ & maximum density of PCM & $20000$ & $\frac{\text{kg}}{\text{m}^{3}}$ \\ -${{ρ_{P}}^{min}}$ & minimum density of PCM & $500$ & $\frac{\text{kg}}{\text{m}^{3}}$ +${{ρ_{P}}^{\text{min}}}$ & minimum density of PCM & $500$ & $\frac{\text{kg}}{\text{m}^{3}}$ \\ -${{ρ_{W}}^{max}}$ & maximum density of water & $1000$ & $\frac{\text{kg}}{\text{m}^{3}}$ +${{ρ_{W}}^{\text{max}}}$ & maximum density of water & $1000$ & $\frac{\text{kg}}{\text{m}^{3}}$ \\ -${{ρ_{W}}^{min}}$ & minimum density of water & $950$ & $\frac{\text{kg}}{\text{m}^{3}}$ +${{ρ_{W}}^{\text{min}}}$ & minimum density of water & $950$ & $\frac{\text{kg}}{\text{m}^{3}}$ \\ \bottomrule \caption{Auxiliary Constants}