SLOPE STABILITY

A. Introduction

1. Force

2. Mass

3. Acceleration

4. Force = mass ´ acceleration

F = m ´ a 

• 1 Newton: unit of force that will impart an acceleration of 1 m/sec/sec to a mass of 1 kg

1 N = 1 kg m/sec/sec

• weight = mass ´ acceleration due to gravity

W = m ´ g   

 

B. Forces Acting on a Block on a Horizontal Surface

1. Forces acting on a block at rest

• normal force = N

• weight = W

• resultant force = R

2. Addition of a force from the side

a. shear force = T

b. angle of internal friction, f

c. resultant force is proportional to:

• normal force, and

• coefficient of friction = tanf

• R = N tanf

d. just before motion occurs:

• T = R

• N = W

• T = R = W tanf

3. Stress: force per unit area = F/A

a. shear stress,    

b. normal stress,    

c. strength: frictional shear strength = S

(1) function of:

• friction, and

• normal force

(2) just before motion occurs, T=R

4. Summary

N = normal force

W = weight; N=W

T = shear force

R = resultant force or friction force

f = angle of internal friction

R = T = N tanf = W tanf

s = normal stress = N/A = W/A

t = shear stress = T/A

S = shear strength = s tanf

 

C. Forces Acting on a Block on an Incline

1. Gravity: resolves into two components

• N = normal force

• T = shear force

• magnitude of N & T are a function of slope angle, q

2. Weight = force due to gravity

• N = W cosq

• T = W sinq

• W = volume ´ unit specific weight of material (kg/m³)

(1) volume = B ´ W_d ´ H

(2) g = unit specific weight

(3) W = B ´ W_d ´ H ´ g

 

B=base                    N=normal force
H=height                 W=weight (force due to gravity)
W_d=width               T=shear force
q =slope angle        R=resultant force

3. Normal stress = s

4. Shear stress = t

5. Shear strength, S = s ´ tanf

 

D. Forces on an Unconsolidated Block

1. Same forces, different calculations

 

a. volume = b ´ h ´ W_d

b. area of contact = B ´ W_d

area of contact =

2. N = W ´ cosq

3. T = W ´ sinq

4. W = b ´ h ´ W_d ´ g

5.  

 

6.  

 

7.  S = s ´ tanf

= h ´ g ´ cos²q ´ tanf

8. Pore pressure and cohesion

a. pore pressure = m

m = g_w x m x h cos²q

g_w = unit specific weight of water

m = fraction of height that’s saturated;

0 £ m £ 1

b. effective normal stress

s' = s - m

c. cohesion: increases shear strength

• S = stanf

• S = c + s'tanf

E. The Factor of Safety: F

F = sum of resisting forces/sum of driving forces

F = shear strength/shear stress  = S/t

F. Factors Leading to Failure

1. Changes in slope gradient

2. Excessive loading

3. Shocks and vibrations

4. Changes in water content

G. Summary