Exercise #05C Slope Stability Calculation

 

 

Objective: To calculate the stability of a slope using a spreadsheet program

 

Instructions: Create a directory entitled username_spreadsheet in your workstation.

 

 

Spreadsheet programs are also very effective in handling calculations that are dependent on many levels of equations. As an example, we will take the calculation of slope stability. Your goal will be to create a spreadsheet that will aid in this calculation. This is also a test of your problem-solving skills, so try to work as independently as possible.

 

Background Information

 

The forces acting on a point along the potential failure plane are illustrated in the figure below. All the variables in the figures, equations and tables are defined in the table beside the figure.

 

 

Force diagram for thin to thick translational slides.

 

 

 

The resisting force of earth materials, whether consolidated bedrock or unconsolidated sediments, is the shear strength (S) of the materials. Shear strength is a combination of forces, including the slope normal component of gravity or normal stress (σ), pore pressure (μ) within the material, which counteracts the normal stress, cohesion of the material (C), and the angle of internal friction (φ). Shear strength is given by the Coulomb Equation:


S = C + (σ - μ)tan&phi                                   (Equation 1)

Normal stress is the vertical component of gravity, resisting downslope movement:


σ = γzcosβcosβ                                          (Equation 2)

The role of water is especially critical in slope stability, but it is incorrect to think of its role as that of lubrication. Water plays a dual role. In increasing the unit weight of material, it increases both the resisting (normal stress) and driving (shear stress) forces. It also creates pore pressure, which opposes the normal stress and therefore reduces the resisting force or shear strength of the material (it is subtracted from normal stress in Equation 1). It is represented by the following equation:


μ = γwmzcosβcosββ                                   (Equation 3)

 

 

Geometry of the vector components of gravity. Unit width of the block is assumed;

block length is infinite in the infinite slope model and is dropped from the equation.


The driving force is shear stress (τ), the slope parallel component of gravity. Shear strength is given by the following equation:

τ = γzcosβsinβ                                   (Equation 4)

Slope stability is typically evaluated in terms of a safety factor, also referred to as a factor of safety and denoted as SF. As it applies to the infinite slope model, the SF is a ratio between resisting and driving forces as shown in Equations 5 & 6:

 

 

It follows that if shear strength is greater than shear stress, then SF > 1 and the slope may be considered stable; if shear strength is less than shear stress, SF <

1 and the slope may be considered unstable. For SF = 1, the slope would be considered in a balanced state, but inherently unstable. In cases where SF ≤ 1,

whether the slope actually fails is another matter as will be discussed later, but the potential for failure is high and mitigation would be warranted.

 

The infinite slope model generally relies on several simplifying assumptions which may cause some limitation to its application. It assumes that:

 

o              failure is the result of translational sliding;

o              the failure plane and water table parallel the ground surface;

o              the failure plane is of infinite length; and

o              failure occurs as a single layer.

 

It also does not account for the impact of adjacent factors like upslope development or downslope modifications of the hillslope or accentuating factors such as ground vibrations or acceleration due to earthquakes.

 

Exercise

 

Given the following values:

 

Cohesion

110 kN/m2

Unit weight of slope material

18 kN/m3

Thickness of slope material above the slide plane

6.3 m

Thickness of saturated slope material above the slide plane

4.4 m

Vertical height of water table above the slide plane

0.37

Slope of the ground surface

38

 

(1)   Create a spreadsheet template (Sheet 1) for calculating for the safety factor of a slope using Equation 6, assuming that the following values will be given to you: cohesion, internal angle of friction, slope of the ground surface, vertical height of water table above the slide plane, unit weight of slope material, and the thickness of the slope material above the slide plane.

 

(2)   Using the template that you created, calculate the safety factor of a hypothetical slope with the values given above.

 

(3)   Is the hypothetical slope safe or not?

 

(4)   Create another spreadsheet template (Sheet 2) for calculating for the safety factor of a slope using Equation 5, and calculate the safety factor of the slope using this spreadsheet. How does the result in this second calculation compare with the first?

 

Answer all questions in a text file to be saved in the same directory as the spreadsheets.

 

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