Contents preface (VII) introduction 1—37

15.4.8. Stability of Foundation
The factor of safety against shear for the foundation material is the ratio of the shear strength to shear stress at a location in the foundation where maximum intensity of shear stress occurs. Since the factor of safety corresponds to the maximum shear stress, its acceptable value should only be slightly greater than unity for the foundation of an embankment dam. The magnitude and location of the maximum shear stress can be determined approximately as follows:
The horizontal shear, S under a slope of the dam is equal to the difference between the lateral thrust on a vertical through the upper end of the slope and a vertical through the toe of the slope. Thus, with reference to Fig. 15.32,

	h1^2	− h2^2		2	F		φ1 I
S = ρg			tan		G 45	−		J	(15.36)
	2
					H		2 K

where, ρ = average mass density of the soil, g = acceleration due to gravity,
φ₁ = equivalent angle of internal friction, determined by the equation ρgh₁ tan φ₁ = c + ρgh₁ tan φ where, c and φ are actual properties of the foundation soil, and h₁ and h₂ are heights, as shown in Fig. 15.32, of the upper and lower ends of the slope, respectively, above a stratum which is much stronger than the overlying foundation material.

h₁

b

h₂

Base of dam

Hard stratum

Fig. 15.32 Shear stress in foundation of embankment
The average shear stress s_a is equal to S/b. Here, b is the horizontal length of the slope. Results of photoelastic investigations indicate that the maximum intenstiy of shear equals 1.4 times the average shear s_a, and occurs at a distance of 0.4 b from the upper end of the slope. One can, alternatively, compare average values of shear stress and shear strength and accept a relatively higher factor of safety for the stability of the foundation of an embankment.