Contents preface (VII) introduction 1—37

15.4.6.2. Wedge (or Sliding Block) Method
This method is used when the slip surface can be approximated by two or three straight lines. Such a situation arises when the slope is underlain by a strong stratum such as rock or there is a weak layer included within or beneath the slope. In such circumstances, an accurate stability analysis can be carried out by dividing the trial sliding mass into two or three blocks of soil and examining the equilibrium of each block. The upper block (or wedge) is called the driving (or active) block and the lower block is called the resisting (or passive) block. In a three-wedge system, the central block is called the sliding block (Fig. 15.30).

EMBANKMENT DAMS				529
	Active	Central	Passive
	wedge	block	wedge
	a	Assumed failure
	c
	^W1	surface
Foundation	PA	W2	W3
e	f
	b			d	PP
(a) Dam on weak	Weak layer (Soft clay or silt or fine
foundation	sand with high pore pressure)
	Driving	Resisting
	wedge	wedge

	Assumed failure
	surface
	Core
(b) Dam on strong	Shell of strong granular
	material (sand, gravel or rock)
foundation

Fig. 15.30 Conditions for applicability of wedge analysis
The factor of safety can be estimated by any of the methods discussed earlier. Alternatively, assuming that the active and passive wedges are at failure and that the total forces on the vertical planes (bc and de) are horizontal, one can estimate the factor of safety as the ratio of the force P₁ available on bd to resist the movement of the central block and the unbalanced force, P_A – P_P . This means,

F =	Cbd + ( W2	− Ubd ) tan φ ′ bd	(15.34)
PA − PP

where, the subscript bd is used for the plane bd and the subscript 2 is for the central block.
Example 15.5 Determine the factor of safety for the slip surface shown in Fig. 15.31 for sudden drawdown condition with the following properties of the embankment material:

1

2	2
1	3
	4
5

6

8 ⁷

Fig. 15.31 Slip surface for Example 15.5

530	IRRIGATION AND WATER RESOURCES ENGINEERING
Saturated weight	= 21.0 kN/m^3
Submerged weight	= 11.0 kN/m^3
Cohesion	= 24.5 kN/m^2
Angle of internal friction, φ′	= 35°

Angle α, arc length, and area of different slices are given in the first four columns of Table 15.3.

Solution: Since pore pressures are not known, the driving force (T-component) and the resisting force ( N-component) are calculated using saturated and submerged weights, respectively, and are shown in Table 15.3.
Table 15.3 Data and solution for Example 15.4

				T-Component	N-Component
Slice	α (degrees)	Arc length	Area of
Weight, W	W sin α	Weight, W		W cos α
		(metres)		slice (m^2)
		(kN/m)			(kN/m)	(kN/m)	(kN/m)
1	54.5	6.70	12.26		257.46	209.60		134.86	78.31
2	41	3.80	19.51		409.71	268.79		214.61	161.97
3	31	3.50	21.37		448.77	231.13	235.07	201.49
4	22	3.35	20.90	438.90	164.42	229.90	213.16
5	13	3.05	19.97	419.37	94.34	219.67	214.04
6	5	3.05	16.72	351.12	30.60	183.92	183.22
7	– 3.5	3.05	12.08	253.68	– 15.49	132.88	132.63
8	– 13	4.30	6.69	140.49	– 31.60	73.59	71.70
		30.80			951.79		1256.52

Using Eq. (15.33)

_{Factor of Safety =} 30.80 × 24.5 + 1256.52 tan 35° _{= 1.72} 95179.
15.4.7. Seismic Considerations in Stability Analysis
Seismic forces reduce the margin of safety of an embankment dam. Therefore, when an embankment dam is located in a seismic region, the stability analysis must also consider earthquake forces. During an earthquake, the ground surface oscillates randomly in different directions. This motion can be represented by horizontal and vertical components. A rigid structure is expected to follow the oscillations of its base in the absence of relative deformation from the base of the structure to its top. The amplitude of the oscillations and the acceleration vary along the height of the structure. An earth dam should be treated as a flexible structure for determining dynamic pressure due to earthquake. However, a simple method to account for earthquake forces in the design of structures is based on seismic coefficients. In this method, basic seismic coefficients or earthquake acceleration coefficients are used. The seismic coefficient is defined as the ratio of earthquake acceleration in a particular direction to the gravitational acceleration. If α_h is the horizontal earthquake acceleration coefficient then the additional inertial force of the soil mass (of the slice under consideration) is taken as α_hW in the horizontal direction. Obviously, a force equal to α_hW cos α is added to the tangential forces and α_hW sin α

EMBANKMENT DAMS

531

is deducted from the forces acting in the normal direction. The factor of safety, therefore, becomes

F =	Σ C + Σ ( N − U − α hW sin α ) tan φ′	(15.35)
Σ (W sin α + α hW cos α)

This simple way of accounting the seismic effects in the stability analysis is based on the pseudostatic concept in which the dynamic effects of an earthquake are replaced by a static force, and in which limit equilibrium is maintained (6).