Software Requirements Specification for Slope Stability analysis Program

Henry Frankis and Brooks MacLachlan

Table of Contents

An outline of all sections included in this SRS is recorded here for easy reference.

• Table of Contents
• Reference Material
  • Table of Units
  • Table of Symbols
  • Abbreviations and Acronyms
• Introduction
  • Purpose of Document
  • Scope of Requirements
  • Characteristics of Intended Reader
  • Organization of Document
• General System Description
  • System Context
  • User Characteristics
  • System Constraints
• Specific System Description
  • Problem Description
    • Terminology and Definitions
    • Physical System Description
    • Goal Statements
  • Solution Characteristics Specification
    • Assumptions
    • Theoretical Models
    • General Definitions
    • Data Definitions
    • Instance Models
    • Data Constraints
    • Properties of a Correct Solution
• Requirements
  • Functional Requirements
  • Non-Functional Requirements
• Likely Changes
• Unlikely Changes
• Traceability Matrices and Graphs
• Values of Auxiliary Constants
• References

Reference Material

This section records information for easy reference.

Table of Units

The unit system used throughout is SI (Système International d’Unités). In addition to the basic units, several derived units are also used. For each unit, the Table of Units lists the symbol, a description, and the SI name.

Table of Units

Table of Symbols

The symbols used in this document are summarized in the Table of Symbols along with their units. Throughout the document, a subscript $i$ indicates that the value will be taken at, and analyzed at, a slice or slice interface composing the total slip mass. For vector quantities, the units shown are for each component of the vector.

Table of Symbols

Abbreviations and Acronyms

Abbreviations and Acronyms

Introduction

A slope of geological mass, composed of soil and rock and sometimes water, is subject to the influence of gravity on the mass. This can cause instability in the form of soil or rock movement. The effects of soil or rock movement can range from inconvenient to seriously hazardous, resulting in significant life and economic losses. Slope stability is of interest both when analysing natural slopes, and when designing an excavated slope. Slope stability analysis is the assessment of the safety of a slope, identifying the surface most likely to experience slip and an index of its relative stability known as the factor of safety..

The following section provides an overview of the Software Requirements Specification (SRS) for a slope stability analysis problem. The developed program will be referred to as the Slope Stability analysis Program (SSP) based on the original, manually created version of SSP. This section explains the purpose of this document, the scope of the requirements, the characteristics of the intended reader, and the organization of the document.

Purpose of Document

The primary purpose of this document is to record the requirements of SSP. Goals, assumptions, theoretical models, definitions, and other model derivation information are specified, allowing the reader to fully understand and verify the purpose and scientific basis of SSP. With the exception of system constraints, this SRS will remain abstract, describing what problem is being solved, but not how to solve it.

This document will be used as a starting point for subsequent development phases, including writing the design specification and the software verification and validation plan. The design document will show how the requirements are to be realized, including decisions on the numerical algorithms and programming environment. The verification and validation plan will show the steps that will be used to increase confidence in the software documentation and the implementation. Although the SRS fits in a series of documents that follow the so-called waterfall model, the actual development process is not constrained in any way. Even when the waterfall model is not followed, as Parnas and Clements point out parnasClements1986, the most logical way to present the documentation is still to “fake” a rational design process.

Scope of Requirements

The scope of the requirements includes stability analysis of a two-dimensional (2D) soil mass, composed of a single homogeneous layer with constant material properties. The soil mass is assumed to extend infinitely in the third dimension. The analysis will be at an instant in time; factors that may change the soil properties over time will not be considered.

Characteristics of Intended Reader

Reviewers of this documentation should have an understanding of undergraduate level 4 physics and undergraduate level 2 or higher solid mechanics. It would be an asset to understand soil mechanics. The users of SSP can have a lower level of expertise, as explained in Sec:User Characteristics.

Organization of Document

The organization of this document follows the template for an SRS for scientific computing software proposed by koothoor2013, smithLai2005, smithEtAl2007, and smithKoothoor2016. The presentation follows the standard pattern of presenting goals, theories, definitions, and assumptions. For readers that would like a more bottom up approach, they can start reading the instance models and trace back to find any additional information they require.

The goal statements are refined to the theoretical models and the theoretical models to the instance models. The instance models provide the set of algebraic equations that must be solved.

General System Description

This section provides general information about the system. It identifies the interfaces between the system and its environment, describes the user characteristics, and lists the system constraints.

System Context

Fig:sysCtxDiag shows the system context. A circle represents an external entity outside the software. A rectangle represents the software system itself (SSP). Arrows are used to show the data flow between the system and its environment.

The responsibilities of the user and the system are as follows:

• User Responsibilities
  • Provide the input data related to the soil layer(s) and water table (if applicable), ensuring conformation to input data format required by SSP
  • Ensure that consistent units are used for input variables
  • Ensure required software assumptions are appropriate for the problem to which the user is applying the software
• SSP Responsibilities
  • Detect data type mismatch, such as a string of characters input instead of a floating point number
  • Verify that the inputs satisfy the required physical and other data constraints
  • Identify the critical slip surface within the possible input range
  • Find the factor of safety for the slope
  • Find the interslice normal force and shear force along the critical slip surface

User Characteristics

The end user of SSP should have an understanding of undergraduate Level 1 Calculus and Physics, and be familiar with soil and material properties, specifically effective cohesion, effective angle of friction, and unit weight.

System Constraints

The Morgenstern-Price method morgenstern1965, which involves dividing the slope into vertical slices, will be used to derive the equations for analysing the slope.

Specific System Description

This section first presents the problem description, which gives a high-level view of the problem to be solved. This is followed by the solution characteristics specification, which presents the assumptions, theories, and definitions that are used.

Problem Description

A system is needed to evaluate the factor of safety of a slope’s slip surface and identify the critical slip surface of the slope, as well as the interslice normal force and shear force along the critical slip surface.

Terminology and Definitions

This subsection provides a list of terms that are used in the subsequent sections and their meaning, with the purpose of reducing ambiguity and making it easier to correctly understand the requirements.

• 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.
• Slip surface: A surface within a slope that has the potential to fail or displace due to load or other forces.
• Critical slip surface: Slip surface of the slope that has the lowest factor of safety, and is therefore most likely to experience failure.
• Water table: The upper boundary of a saturated zone in the ground.
• Stress: The ratio of an applied force to a cross-sectional area.
• Strain: A measure of deformation representing the displacement between particles in the body relative to a reference length.
• Normal force: A force applied perpendicular to the plane of the material.
• Shear force: A force applied parallel to the plane of the material.
• Mobilized shear force: The shear force in the direction of potential motion, thus encouraging motion along the plane.
• Resistive shear force: The shear force in the direction opposite to the direction of potential motion, thus hindering motion along the plane.
• Effective forces and stresses: The normal force or normal stress carried by the soil skeleton, composed of the effective force or stress and the force or stress exerted by water.
• Cohesion: An attractive force between adjacent particles that holds the matter together.
• Isotropy: A condition where the value of a property is independent of the direction in which it is measured.
• Plane strain: A condition where the resultant stresses in one of the directions of a three-dimensional material can be approximated as zero. This condition results when a body is constrained to not deform in one direction, or when the length of one dimension of the body dominates the others, to the point where it can be assumed as infinite. Stresses in the direction of the dominant dimension can be approximated as zero.

Physical System Description

The physical system of SSP, as shown in Fig:PhysicalSystem, includes the following elements:

PS1: A slope comprised of one soil layer.

PS2: A water table, which may or may not exist.

Morgenstern-Price analysis morgenstern1965 of the slope involves representing the slope as a series of vertical slices. As shown in Fig:IndexConvention, the index $i$ is used to denote a value for a single slice, and an interslice value at a given index $i$ refers to the value between slice $i$ and adjacent slice $i+1$.

A free body diagram of the forces acting on a slice is displayed in Fig:ForceDiagram. The specific forces and symbols will be discussed in detail in Sec:General Definitions and Sec:Data Definitions.

Goal Statements

Given the shape of the soil mass, the location of the water table, and the material properties of the soil, the goal statements are:

Identify-Crit-and-FS: Identify the critical slip surface and the corresponding factor of safety.

Determine-Normal-Forces: Determine the interslice normal forces between each pair of vertical slices of the slope.

Determine-Shear-Forces: Determine the interslice shear forces between each pair of vertical slices of the slope.

Solution Characteristics Specification

The instance models that govern SSP are presented in the Instance Model Section. The information to understand the meaning of the instance models and their derivation is also presented, so that the instance models can be verified.

Assumptions

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.

Slip-Surface-Concave: The slip surface is concave with respect to the slope surface. The (${\boldsymbol{x}_{\text{slip}}}$, ${\boldsymbol{y}_{\text{slip}}}$) coordinates of a slip surface follow a concave up function. (RefBy: IM:crtSlpId.)

Factor-of-Safety: The factor of safety is assumed to be constant across the entire slip surface. (RefBy: GD:mobShr.)

Soil-Layer-Homogeneous: The soil mass is homogeneous, with consistent soil properties throughout. (RefBy: GD:sliceWght, GD:resShr, and LC:Calculate-Inhomogeneous-Soil-Layers.)

Soil-Properties: The soil properties are independent of dry or saturated conditions, with the exception of unit weight. (RefBy: GD:resShr.)

Soil-Layers-Isotropic: The soil mass is treated as if the effective cohesion and effective angle of friction are isotropic properties. (RefBy: GD:resShr.)

Interslice-Norm-Shear-Forces-Linear: Following the assumption of Morgenstern and Price (morgenstern1965), interslice normal forces and interslice shear forces have a proportional relationship, depending on a proportionality constant ($λ$) and a function ($\boldsymbol{f}$) describing variation depending on $x$ position. (RefBy: IM:nrmShrFor, GD:normShrR, IM:fctSfty, and UC:Normal-And-Shear-Linear-Only.)

Plane-Strain-Conditions: The slope and slip surface extends far into and out of the geometry ($z$ coordinate). This implies plane strain conditions, making 2D analysis appropriate. (RefBy: GD:srfWtrF, GD:sliceWght, GD:resShr, GD:effNormF, and GD:baseWtrF.)

Effective-Norm-Stress-Large: The effective normal stress is large enough that the shear strength to effective normal stress relationship can be approximated as a linear relationship. (RefBy: TM:equilibrium and UC:2D-Analysis-Only.)

Surface-Base-Slice-between-Interslice-Straight-Lines: The surface and base of a slice are approximated as straight lines. (RefBy: GD:srfWtrF, GD:sliceWght, TM:mcShrStrgth, GD:baseWtrF, DD:slcHeight, DD:angleB, and DD:angleA.)

Edge-Slices: The interslice forces at the 0th and $n$th interslice interfaces are zero. (RefBy: IM:nrmShrFor, IM:intsliceFs, and IM:fctSfty.)

Seismic-Force: There is no seismic force acting on the slope. (RefBy: IM:nrmShrFor, IM:fctSfty, and LC:Calculate-Seismic-Force.)

Surface-Load: There is no imposed surface load, and therefore no external forces, acting on the slope. (RefBy: IM:nrmShrFor, IM:fctSfty, and LC:Calculate-External-Force.)

Water-Intersects-Base-Edge: The water table only intersects the base of a slice at an edge of the slice. (RefBy: GD:sliceWght and GD:baseWtrF.)

Water-Intersects-Surface-Edge: The water table only intersects the slope surface at the edge of a slice. (RefBy: GD:srfWtrF and GD:sliceWght.)

Negligible-Effect-Surface-Slope-Seismic: The effect of the slope of the surface of the soil on the seismic force is assumed to be negligible. (RefBy: GD:momentEql.)

Hydrostatic-Force-Slice-Midpoint: The resultant surface hydrostatic forces act into the midpoint of each slice surface and the resultant base hydrostatic forces act into the midpoint of each slice base. (RefBy: GD:srfWtrF, GD:momentEql, and GD:baseWtrF.)

Theoretical Models

This section focuses on the general equations and laws that SSP is based on.

General Definitions

This section collects the laws and equations that will be used to build the instance models.

Normal force equilibrium is derived from the free body diagram of Fig:ForceDiagram in Sec:Physical System Description.

Base shear force equilibrium is derived from the free body diagram of Fig:ForceDiagram in Sec:Physical System Description.

Derived by substituting DD:normStress and DD:tangStress into the Mohr-Coulomb shear strength, TM:mcShrStrgth, and multiplying both sides of the equation by the area of the slice in the shear-$z$ plane. Since the slope is assumed to extend infinitely in the $z$-direction (A:Plane-Strain-Conditions), the resulting forces are expressed per metre in the $z$-direction. The effective angle of friction $φ’$ and the effective cohesion $c’$ are not indexed by $i$ because they are assumed to be isotropic (A:Soil-Layers-Isotropic) and the soil is assumed to be homogeneous, with constant soil properties throughout (A:Soil-Layer-Homogeneous, A:Soil-Properties).

Mobilized shear forces is derived by dividing the definition of the $\boldsymbol{P}$ from GD:resShr by the definition of the factor of safety from 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 (A:Factor-of-Safety).

Derived by substituting DD:normStress into TM:effStress and multiplying both sides of the equation by the area of the slice in the shear-$z$ plane. Since the slope is assumed to extend infinitely in the $z$-direction (A:Plane-Strain-Conditions), the resulting forces are expressed per metre in the $z$-direction.

Moment is equal to torque, so the equation from DD:torque will be used to calculate moments:

$$\boldsymbol{τ}=\boldsymbol{u}\times\boldsymbol{F}$$

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:

$$M={F_{\text{rot}}}\,r$$

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 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:

$$-{\boldsymbol{G}}_{i}\,\left({\boldsymbol{h}_{\text{z},i}}+\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)$$

For the $i-1$th slice interface, the moment is similar but in the opposite direction:

$${\boldsymbol{G}}_{i-1}\,\left({\boldsymbol{h}_{\text{z},i-1}}-\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)$$

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:

$$-{\boldsymbol{H}}_{i}\,\left(\frac{1}{3}\,{\boldsymbol{h}_{\text{z,w},i}}+\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)$$

The moment for the interslice normal water force acting on slice interface $i-1$ is:

$${\boldsymbol{H}}_{i-1}\,\left(\frac{1}{3}\,{\boldsymbol{h}_{\text{z,w},i-1}}+\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)$$

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:

$${\boldsymbol{X}}_{i}\,\frac{{\boldsymbol{b}}_{i}}{2}$$

The interslice shear force at slice interface $i-1$ also tends to rotate in the clockwise direction, and has the same length of the moment arm, so the moment is:

$${\boldsymbol{X}}_{i-1}\,\frac{{\boldsymbol{b}}_{i}}{2}$$

Seismic forces act over the entire height of the slice. For each horizontal segment of the slice, the seismic force is ${K_{\text{c}}}\,{\boldsymbol{W}}_{i}$ where ${\boldsymbol{W}}_{i}$ can be expressed as $γ\,{\boldsymbol{b}}_{i}\,y$ using GD:weight where $y$ is the height of the segment under consideration. The corresponding length of the moment arm is $y$, the height from the base of the slice to the segment under consideration. In reality, the forces near the surface of the soil mass are slightly different due to the slope of the surface, but this difference is assumed to be negligible (A:Negligible-Effect-Surface-Slope-Seismic). The resultant moment from the forces on all of the segments with an equivalent resultant length of the moment arm is determined by taking the integral over the slice height. The forces tend to rotate in the counterclockwise direction, so the moment is negative:

$$-\int_{0}^{{\boldsymbol{h}}_{i}}{{K_{\text{c}}}\,γ\,{\boldsymbol{b}}_{i}\,y}\,dy$$

Solving the definite integral yields:

$$-{K_{\text{c}}}\,γ\,{\boldsymbol{b}}_{i}\,\frac{{\boldsymbol{h}}_{i}^{2}}{2}$$

Using GD:weight again to express $γ\,{\boldsymbol{b}}_{i}\,{\boldsymbol{h}}_{i}$ as ${\boldsymbol{W}}_{i}$, the moment is:

$$-{K_{\text{c}}}\,{\boldsymbol{W}}_{i}\,\frac{{\boldsymbol{h}}_{i}}{2}$$

The surface hydrostatic force acts into the midpoint of the surface of the slice (A:Hydrostatic-Force-Slice-Midpoint). Thus, the vertical component of the force acts directly towards the point of rotation, and has a moment of zero. The horizontal component of the force tends to rotate in a clockwise direction and the length of the moment arm is the entire height of the slice. Thus, the moment is:

$${\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)\,{\boldsymbol{h}}_{i}$$

The external force again acts into the midpoint of the slice surface, so the vertical component does not contribute to the moment, and the length of the moment arm is again the entire height of the slice. The moment is:

$${\boldsymbol{Q}}_{i}\,\sin\left({\boldsymbol{ω}}_{i}\right)\,{\boldsymbol{h}}_{i}$$

The base hydrostatic force and slice weight both act in the direction of the point of rotation (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:

$$0=-{\boldsymbol{G}}_{i}\,\left({\boldsymbol{h}_{\text{z},i}}+\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+{\boldsymbol{G}}_{i-1}\,\left({\boldsymbol{h}_{\text{z},i-1}}-\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)-{\boldsymbol{H}}_{i}\,\left(\frac{1}{3}\,{\boldsymbol{h}_{\text{z,w},i}}+\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+{\boldsymbol{H}}_{i-1}\,\left(\frac{1}{3}\,{\boldsymbol{h}_{\text{z,w},i-1}}-\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+\frac{{\boldsymbol{b}}_{i}}{2}\,\left({\boldsymbol{X}}_{i}+{\boldsymbol{X}}_{i-1}\right)+\frac{-{K_{\text{c}}}\,{\boldsymbol{W}}_{i}\,{\boldsymbol{h}}_{i}}{2}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)\,{\boldsymbol{h}}_{i}+{\boldsymbol{Q}}_{i}\,\sin\left({\boldsymbol{ω}}_{i}\right)\,{\boldsymbol{h}}_{i}$$

Detailed derivation of weight:

Under the influence of gravity, and assuming a 2D Cartesian coordinate system with down as positive, an object has an acceleration vector of:

$$\boldsymbol{a}\text{(}t\text{)}=\begin{bmatrix}0\\\boldsymbol{g}\,\boldsymbol{\hat{j}}\end{bmatrix}$$

Since there is only one non-zero vector component, the scalar value $W$ will be used for the weight. In this scenario, Newton’s second law of motion from TM:NewtonSecLawMot can be expressed as:

$$W=m\,\boldsymbol{g}$$

Mass can be expressed as density multiplied by volume, resulting in:

$$W=ρ\,V\,\boldsymbol{g}$$

Substituting specific weight as the product of density and gravitational acceleration yields:

$$W=V\,γ$$

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 GD:weight yields:

$${\boldsymbol{W}}_{i}={\boldsymbol{V}_{\text{sat},i}}\,{γ_{\text{sat}}}$$

Due to 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:

$${\boldsymbol{W}}_{i}={\boldsymbol{b}}_{i}\,\frac{1}{2}\,\left({\boldsymbol{y}_{\text{slope},i}}-{\boldsymbol{y}_{\text{slip},i}}+{\boldsymbol{y}_{\text{slope},i-1}}-{\boldsymbol{y}_{\text{slip},i-1}}\right)\,{γ_{\text{sat}}}$$

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 GD:weight yields:

$${\boldsymbol{W}}_{i}={\boldsymbol{V}_{\text{dry},i}}\,{γ_{\text{dry}}}$$

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:

$${\boldsymbol{W}}_{i}={\boldsymbol{b}}_{i}\,\frac{1}{2}\,\left({\boldsymbol{y}_{\text{slope},i}}-{\boldsymbol{y}_{\text{slip},i}}+{\boldsymbol{y}_{\text{slope},i-1}}-{\boldsymbol{y}_{\text{slip},i-1}}\right)\,{γ_{\text{dry}}}$$

For the case where the water table is between the slope surface and slip surface, the weights are the sums of 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 GD:weight and adding them together yields:

$${\boldsymbol{W}}_{i}={\boldsymbol{V}_{\text{dry},i}}\,{γ_{\text{dry}}}+{\boldsymbol{V}_{\text{sat},i}}\,{γ_{\text{sat}}}$$

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 (A:Water-Intersects-Surface-Edge, 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:

$${\boldsymbol{W}}_{i}={\boldsymbol{b}}_{i}\,\frac{1}{2}\,\left(\left({\boldsymbol{y}_{\text{slope},i}}-{\boldsymbol{y}_{\text{wt},i}}+{\boldsymbol{y}_{\text{slope},i-1}}-{\boldsymbol{y}_{\text{wt},i-1}}\right)\,{γ_{\text{dry}}}+\left({\boldsymbol{y}_{\text{wt},i}}-{\boldsymbol{y}_{\text{slip},i}}+{\boldsymbol{y}_{\text{wt},i-1}}-{\boldsymbol{y}_{\text{slip},i-1}}\right)\,{γ_{\text{sat}}}\right)$$

The base hydrostatic forces come from the hydrostatic pressure exerted by the water above the base of each slice. The equation for hydrostatic pressure from GD:hsPressure is:

$$p=γ\,h$$

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 (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 $i-1$:

$$\frac{1}{2}\,\left({\boldsymbol{y}_{\text{wt},i}}-{\boldsymbol{y}_{\text{slip},i}}+{\boldsymbol{y}_{\text{wt},i-1}}-{\boldsymbol{y}_{\text{slip},i-1}}\right)$$

Due to 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 ${\boldsymbol{L}_{b,i}}$, assuming the water table does not intersect a slice base except at a slice edge (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:

$${\boldsymbol{U}_{\text{b},i}}={\boldsymbol{L}_{b,i}}\,{γ_{w}}\,\frac{1}{2}\,\left({\boldsymbol{y}_{\text{wt},i}}-{\boldsymbol{y}_{\text{slip},i}}+{\boldsymbol{y}_{\text{wt},i-1}}-{\boldsymbol{y}_{\text{slip},i-1}}\right)$$

This equation is the non-zero case of 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.

The surface hydrostatic forces come from the hydrostatic pressure exerted by the water above the surface of each slice. The equation for hydrostatic pressure from GD:hsPressure is:

$$p=γ\,h$$

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 (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 $i-1$:

$$\frac{1}{2}\,\left({\boldsymbol{y}_{\text{wt},i}}-{\boldsymbol{y}_{\text{slope},i}}+{\boldsymbol{y}_{\text{wt},i-1}}-{\boldsymbol{y}_{\text{slope},i-1}}\right)$$

Due to 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 ${\boldsymbol{L}_{s,i}}$, assuming the water table does not intersect a slice surface except at a slice edge (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:

$${\boldsymbol{U}_{\text{g},i}}={\boldsymbol{L}_{s,i}}\,{γ_{w}}\,\frac{1}{2}\,\left({\boldsymbol{y}_{\text{wt},i}}-{\boldsymbol{y}_{\text{slope},i}}+{\boldsymbol{y}_{\text{wt},i-1}}-{\boldsymbol{y}_{\text{slope},i-1}}\right)$$

This equation is the non-zero case of 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.

Data Definitions

This section collects and defines all the data needed to build the instance models.

Instance Models

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

The goals GS:Identify-Crit-and-FS, GS:Determine-Normal-Forces, and GS:Determine-Shear-Forces are met by the simultaneous solution of IM:fctSfty, IM:nrmShrFor, and IM:intsliceFs. The goal GS:Identify-Crit-and-FS is also contributed to by IM:crtSlpId.

The Morgenstern-Price method is a vertical slice, limit equilibrium slope stability analysis method. Analysis is performed by breaking the assumed slip surface into a series of vertical slices of mass. Static equilibrium analysis is performed, using two force equations and one moment equation as in TM:equilibrium. The problem is statically indeterminate with only these 3 equations and one constitutive equation (the Mohr Coulomb shear strength of TM:mcShrStrgth) so the assumption GD:normShrR and corresponding equation GD:normShrR are used. The force equilibrium equations can be modified to be expressed only in terms of known physical values, as done in GD:resShearWO and GD:mobShearWO.

The mobilized shear force defined in GD:bsShrFEq can be substituted into the definition of mobilized shear force based on the factor of safety, from GD:mobShr yielding Equation (1) below:

$$\left({\boldsymbol{W}}_{i}-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}+{\boldsymbol{U}_{\text{g},i}}\,\cos\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\cos\left({\boldsymbol{ω}}_{i}\right)\right)\,\sin\left({\boldsymbol{α}}_{i}\right)-\left(-{K_{\text{c}}}\,{\boldsymbol{W}}_{i}-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}-{\boldsymbol{H}}_{i}+{\boldsymbol{H}}_{i-1}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\sin\left({\boldsymbol{ω}}_{i}\right)\right)\,\cos\left({\boldsymbol{α}}_{i}\right)=\frac{{\boldsymbol{N’}}_{i}\,\tan\left(φ’\right)+c’\,{\boldsymbol{L}_{b,i}}}{{F_{\text{S}}}}$$

An expression for the effective normal forces, $\boldsymbol{N’}$, can be derived by substituting the normal forces equilibrium from GD:normForcEq into the definition for effective normal forces from GD:resShearWO. This results in Equation (2):

$${\boldsymbol{N’}}_{i}=\left({\boldsymbol{W}}_{i}-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}+{\boldsymbol{U}_{\text{g},i}}\,\cos\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\cos\left({\boldsymbol{ω}}_{i}\right)\right)\,\cos\left({\boldsymbol{α}}_{i}\right)+\left(-{K_{\text{c}}}\,{\boldsymbol{W}}_{i}-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}-{\boldsymbol{H}}_{i}+{\boldsymbol{H}}_{i-1}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\sin\left({\boldsymbol{ω}}_{i}\right)\right)\,\sin\left({\boldsymbol{α}}_{i}\right)-{\boldsymbol{U}_{\text{b},i}}$$

Substituting Equation (2) into Equation (1) gives:

$$\left({\boldsymbol{W}}_{i}-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}+{\boldsymbol{U}_{\text{g},i}}\,\cos\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\cos\left({\boldsymbol{ω}}_{i}\right)\right)\,\sin\left({\boldsymbol{α}}_{i}\right)-\left(-{K_{\text{c}}}\,{\boldsymbol{W}}_{i}-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}-{\boldsymbol{H}}_{i}+{\boldsymbol{H}}_{i-1}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\sin\left({\boldsymbol{ω}}_{i}\right)\right)\,\cos\left({\boldsymbol{α}}_{i}\right)=\frac{\left(\left({\boldsymbol{W}}_{i}-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}+{\boldsymbol{U}_{\text{g},i}}\,\cos\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\cos\left({\boldsymbol{ω}}_{i}\right)\right)\,\cos\left({\boldsymbol{α}}_{i}\right)+\left(-{K_{\text{c}}}\,{\boldsymbol{W}}_{i}-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}-{\boldsymbol{H}}_{i}+{\boldsymbol{H}}_{i-1}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\sin\left({\boldsymbol{ω}}_{i}\right)\right)\,\sin\left({\boldsymbol{α}}_{i}\right)-{\boldsymbol{U}_{\text{b},i}}\right)\,\tan\left(φ’\right)+c’\,{\boldsymbol{L}_{b,i}}}{{F_{\text{S}}}}$$

Since the interslice shear forces $\boldsymbol{X}$ and interslice normal forces $\boldsymbol{G}$ are unknown, they are separated from the other terms as follows:

$$\left({\boldsymbol{W}}_{i}+{\boldsymbol{U}_{\text{g},i}}\,\cos\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\cos\left({\boldsymbol{ω}}_{i}\right)\right)\,\sin\left({\boldsymbol{α}}_{i}\right)-\left(-{K_{\text{c}}}\,{\boldsymbol{W}}_{i}-{\boldsymbol{H}}_{i}+{\boldsymbol{H}}_{i-1}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\sin\left({\boldsymbol{ω}}_{i}\right)\right)\,\cos\left({\boldsymbol{α}}_{i}\right)-\left(-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}\right)\,\cos\left({\boldsymbol{α}}_{i}\right)+\left(-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}\right)\,\sin\left({\boldsymbol{α}}_{i}\right)=\frac{\left(\left({\boldsymbol{W}}_{i}+{\boldsymbol{U}_{\text{g},i}}\,\cos\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\cos\left({\boldsymbol{ω}}_{i}\right)\right)\,\cos\left({\boldsymbol{α}}_{i}\right)+\left(-{K_{\text{c}}}\,{\boldsymbol{W}}_{i}-{\boldsymbol{H}}_{i}+{\boldsymbol{H}}_{i-1}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\sin\left({\boldsymbol{ω}}_{i}\right)\right)\,\sin\left({\boldsymbol{α}}_{i}\right)+\left(-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}\right)\,\sin\left({\boldsymbol{α}}_{i}\right)+\left(-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}\right)\,\cos\left({\boldsymbol{α}}_{i}\right)-{\boldsymbol{U}_{\text{b},i}}\right)\,\tan\left(φ’\right)+c’\,{\boldsymbol{L}_{b,i}}}{{F_{\text{S}}}}$$

Applying assumptions A:Seismic-Force and A:Surface-Load, which state that the seismic coefficient and the external forces, respectively, are zero, allows for further simplification as shown below:

$$\left({\boldsymbol{W}}_{i}+{\boldsymbol{U}_{\text{g},i}}\,\cos\left({\boldsymbol{β}}_{i}\right)\right)\,\sin\left({\boldsymbol{α}}_{i}\right)-\left(-{\boldsymbol{H}}_{i}+{\boldsymbol{H}}_{i-1}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)\right)\,\cos\left({\boldsymbol{α}}_{i}\right)-\left(-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}\right)\,\cos\left({\boldsymbol{α}}_{i}\right)+\left(-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}\right)\,\sin\left({\boldsymbol{α}}_{i}\right)=\frac{\left(\left({\boldsymbol{W}}_{i}+{\boldsymbol{U}_{\text{g},i}}\,\cos\left({\boldsymbol{β}}_{i}\right)\right)\,\cos\left({\boldsymbol{α}}_{i}\right)+\left(-{\boldsymbol{H}}_{i}+{\boldsymbol{H}}_{i-1}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)\right)\,\sin\left({\boldsymbol{α}}_{i}\right)+\left(-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}\right)\,\sin\left({\boldsymbol{α}}_{i}\right)+\left(-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}\right)\,\cos\left({\boldsymbol{α}}_{i}\right)-{\boldsymbol{U}_{\text{b},i}}\right)\,\tan\left(φ’\right)+c’\,{\boldsymbol{L}_{b,i}}}{{F_{\text{S}}}}$$

The definitions of GD:resShearWO and GD:mobShearWO are present in this equation, and thus can be replaced by ${\boldsymbol{R}}_{i}$ and ${\boldsymbol{T}}_{i}$, respectively:

$${\boldsymbol{T}}_{i}+\left(-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}\right)\,\sin\left({\boldsymbol{α}}_{i}\right)-\left(-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}\right)\,\cos\left({\boldsymbol{α}}_{i}\right)=\frac{{\boldsymbol{R}}_{i}+\left(\left(-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}\right)\,\cos\left({\boldsymbol{α}}_{i}\right)+\left(-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}\right)\,\sin\left({\boldsymbol{α}}_{i}\right)\right)\,\tan\left(φ’\right)}{{F_{\text{S}}}}$$

The interslice shear forces $\boldsymbol{X}$ can be expressed in terms of the interslice normal forces $\boldsymbol{G}$ using A:Interslice-Norm-Shear-Forces-Linear and GD:normShrR, resulting in:

$${\boldsymbol{T}}_{i}+\left(-λ\,{\boldsymbol{f}}_{i-1}\,{\boldsymbol{G}}_{i-1}+λ\,{\boldsymbol{f}}_{i}\,{\boldsymbol{G}}_{i}\right)\,\sin\left({\boldsymbol{α}}_{i}\right)-\left(-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}\right)\,\cos\left({\boldsymbol{α}}_{i}\right)=\frac{{\boldsymbol{R}}_{i}+\left(\left(-λ\,{\boldsymbol{f}}_{i-1}\,{\boldsymbol{G}}_{i-1}+λ\,{\boldsymbol{f}}_{i}\,{\boldsymbol{G}}_{i}\right)\,\cos\left({\boldsymbol{α}}_{i}\right)+\left(-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}\right)\,\sin\left({\boldsymbol{α}}_{i}\right)\right)\,\tan\left(φ’\right)}{{F_{\text{S}}}}$$

Rearranging yields the following:

$${\boldsymbol{G}}_{i}\,\left(\left(λ\,{\boldsymbol{f}}_{i}\,\cos\left({\boldsymbol{α}}_{i}\right)-\sin\left({\boldsymbol{α}}_{i}\right)\right)\,\tan\left(φ’\right)-\left(λ\,{\boldsymbol{f}}_{i}\,\sin\left({\boldsymbol{α}}_{i}\right)+\cos\left({\boldsymbol{α}}_{i}\right)\right)\,{F_{\text{S}}}\right)={\boldsymbol{G}}_{i-1}\,\left(\left(λ\,{\boldsymbol{f}}_{i-1}\,\cos\left({\boldsymbol{α}}_{i}\right)-\sin\left({\boldsymbol{α}}_{i}\right)\right)\,\tan\left(φ’\right)-\left(λ\,{\boldsymbol{f}}_{i-1}\,\sin\left({\boldsymbol{α}}_{i}\right)+\cos\left({\boldsymbol{α}}_{i}\right)\right)\,{F_{\text{S}}}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{i}-{\boldsymbol{R}}_{i}$$

The definitions for $\boldsymbol{Φ}$ and $\boldsymbol{Ψ}$ from DD:convertFunc1 and DD:convertFunc2 simplify the above to Equation (3):

$${\boldsymbol{G}}_{i}\,{\boldsymbol{Φ}}_{i}={\boldsymbol{Ψ}}_{i-1}\,{\boldsymbol{G}}_{i-1}\,{\boldsymbol{Φ}}_{i-1}+{F_{\text{S}}}\,{\boldsymbol{T}}_{i}-{\boldsymbol{R}}_{i}$$

Versions of Equation (3) instantiated for slices 1 to $n$ are shown below:

$${\boldsymbol{G}}_{1}\,{\boldsymbol{Φ}}_{1}={\boldsymbol{Ψ}}_{0}\,{\boldsymbol{G}}_{0}\,{\boldsymbol{Φ}}_{0}+{F_{\text{S}}}\,{\boldsymbol{T}}_{1}-{\boldsymbol{R}}_{1}$$

$${\boldsymbol{G}}_{2}\,{\boldsymbol{Φ}}_{2}={\boldsymbol{Ψ}}_{1}\,{\boldsymbol{G}}_{1}\,{\boldsymbol{Φ}}_{1}+{F_{\text{S}}}\,{\boldsymbol{T}}_{2}-{\boldsymbol{R}}_{2}$$

$${\boldsymbol{G}}_{3}\,{\boldsymbol{Φ}}_{3}={\boldsymbol{Ψ}}_{2}\,{\boldsymbol{G}}_{2}\,{\boldsymbol{Φ}}_{2}+{F_{\text{S}}}\,{\boldsymbol{T}}_{3}-{\boldsymbol{R}}_{3}$$

…

$${\boldsymbol{G}}_{n-2}\,{\boldsymbol{Φ}}_{n-2}={\boldsymbol{Ψ}}_{n-3}\,{\boldsymbol{G}}_{n-3}\,{\boldsymbol{Φ}}_{n-3}+{F_{\text{S}}}\,{\boldsymbol{T}}_{n-2}-{\boldsymbol{R}}_{n-2}$$

$${\boldsymbol{G}}_{n-1}\,{\boldsymbol{Φ}}_{n-1}={\boldsymbol{Ψ}}_{n-2}\,{\boldsymbol{G}}_{n-2}\,{\boldsymbol{Φ}}_{n-2}+{F_{\text{S}}}\,{\boldsymbol{T}}_{n-1}-{\boldsymbol{R}}_{n-1}$$

$${\boldsymbol{G}}_{n}\,{\boldsymbol{Φ}}_{n}={\boldsymbol{Ψ}}_{n-1}\,{\boldsymbol{G}}_{n-1}\,{\boldsymbol{Φ}}_{n-1}+{F_{\text{S}}}\,{\boldsymbol{T}}_{n}-{\boldsymbol{R}}_{n}$$

Applying A:Edge-Slices, which says that ${\boldsymbol{G}}_{0}$ and ${\boldsymbol{G}}_{n}$ are zero, results in the following special cases: Equation (8) for the first slice:

$${\boldsymbol{G}}_{1}\,{\boldsymbol{Φ}}_{1}={F_{\text{S}}}\,{\boldsymbol{T}}_{1}-{\boldsymbol{R}}_{1}$$

and Equation (9) for the $n$th slice:

$$-\left(\frac{{F_{\text{S}}}\,{\boldsymbol{T}}_{n}-{\boldsymbol{R}}_{n}}{{\boldsymbol{Ψ}}_{n-1}}\right)={\boldsymbol{G}}_{n-1}\,{\boldsymbol{Φ}}_{n-1}$$

Substituting Equation (8) into Equation (4) yields Equation (10):

$${\boldsymbol{G}}_{2}\,{\boldsymbol{Φ}}_{2}={\boldsymbol{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\boldsymbol{T}}_{1}-{\boldsymbol{R}}_{1}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{2}-{\boldsymbol{R}}_{2}$$

which can be substituted into Equation (5) to get Equation (11):

$${\boldsymbol{G}}_{3}\,{\boldsymbol{Φ}}_{3}={\boldsymbol{Ψ}}_{2}\,\left({\boldsymbol{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\boldsymbol{T}}_{1}-{\boldsymbol{R}}_{1}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{2}-{\boldsymbol{R}}_{2}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{3}-{\boldsymbol{R}}_{3}$$

and so on until Equation (12) is obtained from Equation (7):

$${\boldsymbol{G}}_{n-1}\,{\boldsymbol{Φ}}_{n-1}={\boldsymbol{Ψ}}_{n-2}\,\left({\boldsymbol{Ψ}}_{n-3}\,\left({\boldsymbol{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\boldsymbol{T}}_{1}-{\boldsymbol{R}}_{1}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{2}-{\boldsymbol{R}}_{2}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{n-2}-{\boldsymbol{R}}_{n-2}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{n-1}-{\boldsymbol{R}}_{n-1}$$

Equation (9) can then be substituted into the left-hand side of Equation (12), resulting in:

$$-\left(\frac{{F_{\text{S}}}\,{\boldsymbol{T}}_{n}-{\boldsymbol{R}}_{n}}{{\boldsymbol{Ψ}}_{n-1}}\right)={\boldsymbol{Ψ}}_{n-2}\,\left({\boldsymbol{Ψ}}_{n-3}\,\left({\boldsymbol{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\boldsymbol{T}}_{1}-{\boldsymbol{R}}_{1}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{2}-{\boldsymbol{R}}_{2}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{n-2}-{\boldsymbol{R}}_{n-2}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{n-1}-{\boldsymbol{R}}_{n-1}$$

This can be rearranged by multiplying both sides by ${\boldsymbol{Ψ}}_{n-1}$ and then distributing the multiplication of each $\boldsymbol{Ψ}$ over addition to obtain:

$$-\left({F_{\text{S}}}\,{\boldsymbol{T}}_{n}-{\boldsymbol{R}}_{n}\right)={\boldsymbol{Ψ}}_{n-1}\,{\boldsymbol{Ψ}}_{n-2}\,{\boldsymbol{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\boldsymbol{T}}_{1}-{\boldsymbol{R}}_{1}\right)+{\boldsymbol{Ψ}}_{n-1}\,{\boldsymbol{Ψ}}_{n-2}\,{\boldsymbol{Ψ}}_{2}\,\left({F_{\text{S}}}\,{\boldsymbol{T}}_{2}-{\boldsymbol{R}}_{2}\right)+{\boldsymbol{Ψ}}_{n-1}\,\left({F_{\text{S}}}\,{\boldsymbol{T}}_{n-1}-{\boldsymbol{R}}_{n-1}\right)$$

The multiplication of the $\boldsymbol{Ψ}$ terms can be further distributed over the subtractions, resulting in the equation having terms that each either contain an $\boldsymbol{R}$ or a $\boldsymbol{T}$. The equation can then be rearranged so terms containing an $\boldsymbol{R}$ are on one side of the equality, and terms containing a $\boldsymbol{T}$ are on the other. The multiplication by the factor of safety is common to all of the $\boldsymbol{T}$ terms, and thus can be factored out, resulting in:

$${F_{\text{S}}}\,\left({\boldsymbol{Ψ}}_{n-1}\,{\boldsymbol{Ψ}}_{n-2}\,{\boldsymbol{Ψ}}_{1}\,{\boldsymbol{T}}_{1}+{\boldsymbol{Ψ}}_{n-1}\,{\boldsymbol{Ψ}}_{n-2}\,{\boldsymbol{Ψ}}_{2}\,{\boldsymbol{T}}_{2}+{\boldsymbol{Ψ}}_{n-1}\,{\boldsymbol{T}}_{n-1}+{\boldsymbol{T}}_{n}\right)={\boldsymbol{Ψ}}_{n-1}\,{\boldsymbol{Ψ}}_{n-2}\,{\boldsymbol{Ψ}}_{1}\,{\boldsymbol{R}}_{1}+{\boldsymbol{Ψ}}_{n-1}\,{\boldsymbol{Ψ}}_{n-2}\,{\boldsymbol{Ψ}}_{2}\,{\boldsymbol{R}}_{2}+{\boldsymbol{Ψ}}_{n-1}\,{\boldsymbol{R}}_{n-1}+{\boldsymbol{R}}_{n}$$

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 IM:fctSfty:

$${F_{\text{S}}}=\frac{\displaystyle\sum_{i=1}^{n-1}{{\boldsymbol{R}}_{i}\,\displaystyle\prod_{v=i}^{n-1}{{\boldsymbol{Ψ}}_{v}}}+{\boldsymbol{R}}_{n}}{\displaystyle\sum_{i=1}^{n-1}{{\boldsymbol{T}}_{i}\,\displaystyle\prod_{v=i}^{n-1}{{\boldsymbol{Ψ}}_{v}}}+{\boldsymbol{T}}_{n}}$$

${F_{\text{S}}}$ depends on the unknowns $λ$ (IM:nrmShrFor) and $\boldsymbol{G}$ (IM:intsliceFs).

From the moment equilibrium of GD:momentEql with the primary assumption for the Morgenstern-Price method of A:Interslice-Norm-Shear-Forces-Linear and associated definition GD:normShrR, Equation (14) can be derived:

$$0=-{\boldsymbol{G}}_{i}\,\left({\boldsymbol{h}_{\text{z},i}}+\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+{\boldsymbol{G}}_{i-1}\,\left({\boldsymbol{h}_{\text{z},i-1}}-\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)-{\boldsymbol{H}}_{i}\,\left(\frac{1}{3}\,{\boldsymbol{h}_{\text{z,w},i}}+\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+{\boldsymbol{H}}_{i-1}\,\left(\frac{1}{3}\,{\boldsymbol{h}_{\text{z,w},i-1}}-\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+λ\,\frac{{\boldsymbol{b}}_{i}}{2}\,\left({\boldsymbol{G}}_{i}\,{\boldsymbol{f}}_{i}+{\boldsymbol{G}}_{i-1}\,{\boldsymbol{f}}_{i-1}\right)+\frac{-{K_{\text{c}}}\,{\boldsymbol{W}}_{i}\,{\boldsymbol{h}}_{i}}{2}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)\,{\boldsymbol{h}}_{i}+{\boldsymbol{Q}}_{i}\,\sin\left({\boldsymbol{ω}}_{i}\right)\,{\boldsymbol{h}}_{i}$$

Rearranging the equation in terms of $λ$ leads to Equation (15):

$$λ=\frac{-{\boldsymbol{G}}_{i}\,\left({\boldsymbol{h}_{\text{z},i}}+\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+{\boldsymbol{G}}_{i-1}\,\left({\boldsymbol{h}_{\text{z},i-1}}-\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)-{\boldsymbol{H}}_{i}\,\left(\frac{1}{3}\,{\boldsymbol{h}_{\text{z,w},i}}+\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+{\boldsymbol{H}}_{i-1}\,\left(\frac{1}{3}\,{\boldsymbol{h}_{\text{z,w},i-1}}-\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+\frac{-{K_{\text{c}}}\,{\boldsymbol{W}}_{i}\,{\boldsymbol{h}}_{i}}{2}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)\,{\boldsymbol{h}}_{i}+{\boldsymbol{Q}}_{i}\,\sin\left({\boldsymbol{ω}}_{i}\right)\,{\boldsymbol{h}}_{i}}{-\frac{{\boldsymbol{b}}_{i}}{2}\,\left({\boldsymbol{G}}_{i}\,{\boldsymbol{f}}_{i}+{\boldsymbol{G}}_{i-1}\,{\boldsymbol{f}}_{i-1}\right)}$$

This equation can be simplified by applying assumptions A:Seismic-Force and A:Surface-Load, which state that the seismic and external forces, respectively, are zero:

$$λ=\frac{-{\boldsymbol{G}}_{i}\,\left({\boldsymbol{h}_{\text{z},i}}+\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+{\boldsymbol{G}}_{i-1}\,\left({\boldsymbol{h}_{\text{z},i-1}}-\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)-{\boldsymbol{H}}_{i}\,\left(\frac{1}{3}\,{\boldsymbol{h}_{\text{z,w},i}}+\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+{\boldsymbol{H}}_{i-1}\,\left(\frac{1}{3}\,{\boldsymbol{h}_{\text{z,w},i-1}}-\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)\,{\boldsymbol{h}}_{i}}{-\frac{{\boldsymbol{b}}_{i}}{2}\,\left({\boldsymbol{G}}_{i}\,{\boldsymbol{f}}_{i}+{\boldsymbol{G}}_{i-1}\,{\boldsymbol{f}}_{i-1}\right)}$$

Taking the summation of all slices, and applying A:Edge-Slices to set ${\boldsymbol{G}}_{0}$, ${\boldsymbol{G}}_{n}$, ${\boldsymbol{H}}_{0}$, and ${\boldsymbol{H}}_{n}$ equal to zero, a general equation for the proportionality constant $λ$ is developed in Equation (16), which combines IM:nrmShrFor, IM:nrmShrForNum, and IM:nrmShrForDen:

$$λ=\frac{\displaystyle\sum_{i=1}^{n}{{\boldsymbol{b}}_{i}\,\left({{\boldsymbol{F}_{\text{x}}}^{\text{G}}}+{{\boldsymbol{F}_{\text{x}}}^{\text{H}}}\right)\,\tan\left({\boldsymbol{α}}_{i}\right)+{\boldsymbol{h}}_{i}\,-2\,{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)}}{\displaystyle\sum_{i=1}^{n}{{\boldsymbol{b}}_{i}\,\left({\boldsymbol{G}}_{i}\,{\boldsymbol{f}}_{i}+{\boldsymbol{G}}_{i-1}\,{\boldsymbol{f}}_{i-1}\right)}}$$

Equation (16) for $λ$ is a function of the unknown interslice normal forces $\boldsymbol{G}$ (IM:intsliceFs) which itself depends on the unknown factor of safety ${F_{\text{S}}}$ (IM:fctSfty).

See IM:nrmShrFor for the derivation of ${\boldsymbol{C}_{\text{num}}}$.

See IM:nrmShrFor for the derivation of ${\boldsymbol{C}_{\text{den}}}$.

This derivation is identical to the derivation for IM:fctSfty up until Equation (3) shown again below:

$${\boldsymbol{G}}_{i}\,{\boldsymbol{Φ}}_{i}={\boldsymbol{Ψ}}_{i-1}\,{\boldsymbol{G}}_{i-1}\,{\boldsymbol{Φ}}_{i-1}+{F_{\text{S}}}\,{\boldsymbol{T}}_{i}-{\boldsymbol{R}}_{i}$$

A simple rearrangement of Equation (3) leads to Equation (17), also seen in IM:intsliceFs:

$${\boldsymbol{G}}_{i}=\frac{{\boldsymbol{Ψ}}_{i-1}\,{\boldsymbol{G}}_{i-1}+{F_{\text{S}}}\,{\boldsymbol{T}}_{i}-{\boldsymbol{R}}_{i}}{{\boldsymbol{Φ}}_{i}}$$

The cases shown in IM:intsliceFs for when $i=0$, $i=1$, or $i=n$ are derived by applying A:Edge-Slices, which says that ${\boldsymbol{G}}_{0}$ and ${\boldsymbol{G}}_{n}$ are zero, to Equation (17). $\boldsymbol{G}$ depends on the unknowns ${F_{\text{S}}}$ (IM:fctSfty) and $λ$ (IM:nrmShrFor).

Data Constraints

The Data Constraints Table 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.

Input Data Constraints

Properties of a Correct Solution

The Data Constraints Table shows the data constraints on the output variables. The column for physical constraints gives the physical limitations on the range of values that can be taken by the variable.

Output Data Constraints

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.

Functional Requirements

This section provides the functional requirements, the tasks and behaviours that the software is expected to complete.

Read-and-Store: Read the inputs, shown in the table Required Inputs, and store the data.

Verify-Input: Verify that the input data lie within the physical constraints.

Determine-Critical-Slip-Surface: Determine the critical slip surface for the input slope, corresponding to the minimum factor of safety, by using IM:fctSfty, IM:nrmShrFor, and IM:intsliceFs to calculate the factor of safety for a slip surface and using IM:crtSlpId to find the slip surface that minimizes it.

Verify-Output: Verify that the minimum factor of safety and critical slip surface satisfy the physical constraints shown in Properties of a Correct Solution.

Display-Input: Display as output the user-supplied inputs listed in Tab:inputsToOutputTable.

Display-Graph: Display the critical slip surface of the 2D slope, as determined from IM:crtSlpId, graphically.

Display-Factor-of-Safety: Display the value of the factor of safety for the critical slip surface, as determined from IM:fctSfty, IM:nrmShrFor, and IM:intsliceFs.

Display-Interslice-Normal-Forces: Using IM:fctSfty, IM:nrmShrFor, and IM:intsliceFs, calculate and graphically display the interslice normal forces.

Display-Interslice-Shear-Forces: Using IM:fctSfty, IM:nrmShrFor, and IM:intsliceFs, calculate and graphically display the interslice shear forces.

Write-Results-To-File: Provide the option of writing the output result data, as given in FR:Display-Input, FR:Display-Graph, FR:Display-Factor-of-Safety, FR:Display-Interslice-Normal-Forces, and FR:Display-Interslice-Shear-Forces, to a file.

Required Inputs

Inputs to be Returned as Output

Non-Functional Requirements

This section provides the non-functional requirements, the qualities that the software is expected to exhibit.

Correctness: The outputs of the code have the properties of a correct solution.

Understandability: The code is modularized with complete module guide and module interface specification.

Reusability: The code is modularized.

Maintainability: If a likely change is made to the finished software, it will take at most 10$\%$ of the original development time, assuming the same development resources are available.

Likely Changes

This section lists the likely changes to be made to the software.

Calculate-Inhomogeneous-Soil-Layers: A:Soil-Layer-Homogeneous - The system currently assumes the soil mass is homogeneous. In the future, calculations can be added for inconsistent soil properties throughout.

Calculate-Seismic-Force: A:Seismic-Force - The system currently assumes no seismic force. In the future, calculations can be added for the presence of seismic force.

Calculate-External-Force: A:Surface-Load - The system currently assumes no external forces. In the future, calculations can be added for an imposed surface load on the slope.

Unlikely Changes

This section lists the unlikely changes to be made to the software.

Normal-And-Shear-Linear-Only: Changes related to A:Interslice-Norm-Shear-Forces-Linear are not possible due to the dependency of the calculations on the linear relationship between interslice normal forces and interslice shear forces.

2D-Analysis-Only: A:Effective-Norm-Stress-Large allows for 2D analysis with these models only because stress along the $z$-direction is zero. These models do not take into account stress in the $z$-direction, and therefore cannot be used without manipulation to attempt three-dimensional analysis.

Traceability Matrices and Graphs

The purpose of the traceability matrices is to provide easy references on what has to be additionally modified if a certain component is changed. Every time a component is changed, the items in the column of that component that are marked with an “X” should be modified as well. Tab:TraceMatAvsA shows the dependencies of the assumptions on each other. Tab:TraceMatAvsAll shows the dependencies of the data definitions, theoretical models, general definitions, instance models, requirements, likely changes, and unlikely changes on the assumptions. Tab:TraceMatRefvsRef shows the dependencies of the data definitions, theoretical models, general definitions, and instance models on each other. Tab:TraceMatAllvsR shows the dependencies of the requirements and goal statements on the data definitions, theoretical models, general definitions, and instance models.

Traceability Matrix Showing the Connections Between Assumptions and Other Assumptions

Traceability Matrix Showing the Connections Between Assumptions and Other Items

Traceability Matrix Showing the Connections Between Items and Other Sections